Progress report on methods of analysis applicable to monocoque aircraft structures (Aircraft Branch report)

              r
'.
1.v% 7. JI J/
W rt 7, 1l( ~ ! ~1713
INDUS. &. SCI.
File 629.13/Un3as AIR CORPS TECHNICAL REPORT No. 43/3
AIR CORPS INFORMATION CIRCULAR
PUBLISHED BY THE CHIEF OF THE AIR CORPS, WASHINGTON, D. C.
Vol. VIII May 15, 1939 No. 713
PROGRESS REPORT ON METHODS OF ANALYSIS
APPLICABLE TO
MONOCOQUE AIRCRAFT STRUCTURES
(AIRCRAFT BRANCH REPORT)
UNITED STATES
GOVERNMENT PRINTING OFFICE
WASHINGTON: 1940
NON·
Ralph Brown Draughon
LIBRARY
JUN 19 2013
Non•Depoitory
Auburn University
-~
TABLE OF CONTENTS
Purpose and scope _____ _____________________________________________________________________ _
Results and conclusions ________________________ __ ______________ __________________ _______ ____ _ Outline of structural problems _____ __ __ __________________ ___ __ ____ ______________ __ __ __ ________ _
SECTION 1. Critical loads on smooth unstiffened sheet_ _______ _______ ____________ ___ ______ _______ _
2. Isotropic fiat rectangular plates-Compressive loads _ ____ __________ ___ __ ____ __________ _
3. Isotropic curved rectangular plates-Compressive loads _______________ ________________ _
Sechler' s Method __________________ ___ ___ ____________________________________ _
Empirical Method __ ____________________________ ___ _____________ ___ _______ __ _ Other methods __________________________________________________________ __ __ _
4. Stiffener design _______ ______________________ ______ _____ _______ ___ ___ __________ ___ _
5. Determination of crushing strength-Stiffeners __________________________ ____ ________ _
6. The stability of open sections __ ____ ___ _________ __ __________ ___ __ _______ ________ ___ _
7. Column strength-Stiffeners _____ ___ ___ ______ ___ __________________________________ _
8. Notes on stiffener design ___ ___ __ _________________________________________________ _
9. Stiffened fiat plates in compression _____ ____ ______ ___ ___________________ ____________ _
Empirical methods ___________ ___ __ __________ __ ____ ___________ __ ____________ __ Theoretical methods _______________ ___ - -- ---r---- -- --- - -- -- - -- - -- - - - - - - -- -- - - - -
·10. Stiffened curved sheets in compression __ ___________ _____________________________ ____ _
• , Empirical methods ___________ : ____ ___ _____ ___ ______________________________ __ 11. Cylinders in bending __ ______________________ __ __________ _____ _______________ __ ___ _
12. Stiffened cylinders ____ ___ _______ __________ ___ ____ ______________________________ __ 13. Isotropic fiat rectangular plates-Unstiffened-Shear in plane of sheet _____ ____ __ ____ __ .
14. Isotropic fiat rectangular plates-Stiffened-Shear in plane of sheet ______ _____ __ __ __ __ _ _
15. Isotropic curved rectangular plates- Shear on stiffened sheet_ _________ _________ _______ _
16. Corrugated sheet-Axial compressive load parallel to corrugations ____ _____ __________ __ _
17. Corrugated sheet-Column stress __ ____________________________ . __________ __________ _
18. Corrugated sheet-Fixity coefficients ___ ____ __ ___ __ _________________________________ _
19. Effect of curving corrugated sheet __________ ______________________________ __________ _
20. Compression on combined fiat and corrugated panels ___________ ________ ________ ______ _
21. Shear loads parallel and perpendicular to the corrugations ____________________ ___ ___ __ 22. Shear in combined smooth and corrugated panels ________ _____________________ __ __ __ _ _
23. Shear on cylinders-Combined smooth and corrugated sheet_ _________________________ _
(II)
Page
1
1
1
3
3
7
8
8
8
9
11
16
17
20
20
20
25
25
25
27
27
34
34
37
37
39
40
40
40
41
44
45
PROGRESS REPORT ON METHODS OF ANALYSIS APPLI~
CABLE TO MONOCOQUE AIRCRAFT STRUCTURES
(Prepared by J. S. Newell and J. H. Harrington ; coordinated by Capt. P.H. Kemmer, A. C., Materiel Divi­sion,
Air Corps, Wright Field, Dayton, Ohio, May 4, 1937)
PURPOSE AND SCOPE
The study of methods of analysis applicable to the
design of monocoque aircraft structures, the partial
results of which are embodied in this report, was
initiated for the purpose of surveying the rational
theoretical procedures developed for, or applicable to,
stiffened and unstiffened stressed skin structures in the
light of existing test results to see which methods
could be used in aircraft design, which could be used
if modified by the introduction of empirical coefficients,
and which appeared to be inapplicable regardless of
how they were changed.
Efforts have been concentrated on developing ra­tional
or empirical procedures for predicting allowable
stresses on smooth and corrugated sheet under com­pressive
and shear loads with a view toward preparing a
handbook presenting those methods found to be
satisfactory. In some cases where procedures have
been developed and descriptions of them published,
only references have been given. In other cases where
the method is of frequent use in design, an attempt has
been made to summarize the more important items
and to include the more useful curves or data in this
report.
RESULTS AND CONCLUSIONS
In the time available for conducting this study it
has been impossible to consider all the problems
involved in the design of monocoque structures. An
outline was prepared covering the various problems,
and studies were then made to determine those on
which theory and test data were available and on
which theory and tests agreed. In this report methods
are given for predicting allowable compressive stresses
on corrugated sheets of aluminum alloy or stainless
steel and on flat and curved smooth sheet, stiffened
and unstiffened. Procedures which are in reasonable
accord with test data have been developed for predict­ing
the compressive strength of stiffeners and stiffened
sheet and these processes are extended to cover the
design of fuselages and wings. Due to the scarcity of
data against which to check some of the methods pro­posed,
it is necessary to consider them as possible
procedures rather than as established.
Borne data are given for the strength of sheets in
shear but further tests are needed on corrugated sheet
and on curved sheet before satisfactory procedures
(1)
may be developed for design. Further investigations of
stiffened sheets in shear are necessary to evolve methods
for handling this very important problem.
It is concluded that work should be continued on a
classification of problems and on investigations of
theoretical procedures for solving them. The publica­tion
of Timoshenko's "Theory of Elastic Stability" pro_
vides an excellent source book for methods applicable
to thin sheet structures and it is recommended that
more of the procedures described there, and in other
publications, be analyzed and checked against test
data with a view toward extending the ground covered
by this report.
It is believed desirable that lat~· progress reports
should include such matters as rivet and spot welding
practice, methods for inhibiting corrosion in various
types of structures with various materials and, finally,
as many weight data on monocoque structures as can
be obtained. It is fully appreciated that weight data
form part of a manufacturer's engineering investment
and, since they affect the estimated performance of his
airplane, have a direct bearing on his success with ex­perimental
designs, hence constitute one of the items
upon which he competes. It is believed, however,
that the data may be presented so that it will serve as a
guide and check to all, yet divulge specific data on no
one design.
OUTLINE OF STRUCTURAL PROBLEMS
An outline of the basic problems in the analysis of
monocoque aircraft follows. Items considered in this
report are mentioned with references to the pages in­volved.
Items covered in other publications have
references to the articles in question and items upon
which few or no data exist are listed in that way. The
fact that reference is made to any article or theory does
not indicate that the data or methods are approved for
use in airplanes submitted to the Air Corps.
In compiling the data in this report no effort has been
made to follow the order of the outline but related
material has been arranged in what, it is hoped, will
prove to be a useful sequence.
In addition to the references in this outline attention
should be given those in N. A. C. A. Technical Memo­randum
No. 785, "Methods and Formulas for Calculating
the Strength of Plate and Shell Constructions used in
Airplane Design," Heck and Ebner.
I. FLAT PLATES, RECTANGULAR
A. Isotropic.
1. Compression in plane of sheet. Stress to produce
buckling, p. 3. Stress at maximum load, pp. 3-7.
2. Shear in plane of sheet. Stress to produce buck­ling,
p. 34. Stress at maximum load, p. 34.
3. Load normal to plane of sheet. See Timoshenko's
"Strength of Materials," vol. 2, p. 476. Timoshenko's
"Theory of Elastic Stability," ch. VI.
4. Combinations of above. See Timoshenko's
"Theory of Elastic Stability," pp. 350 and 362. Also
Wagner, in W. G. L. Yearbook, 1928, pp. 113-125.
Also Stein, in Stahlbau, Bd. 7 (1934) No. 8, pp. 57-60.
B. Corrugated.
1. Compression parallel to corrugations. Critical
stresses, pp. 37, 38, 39, and 40.
2. Shear in plane of sheet. Critical stresses, pp. 41,
43, 44, and 45.
3. Load normal to plane of sheet. No known data
in aircraft sizes.
4. Combinations of above. No known theory or
tests.
C. Stiffened
1. Compression parallel to stiffeners. Stress at
maximum load, pp. 20-25.
2. Shear in plan.e of sheet. Stress at buckling, pp.
34-36. Stress at maximum load, pp. 36-37.
3. Loads normal to plane of sheet. See references
for isotropic sheet.
4. Combinations of above. No known data.
D. Combinations of smooth and corrugated sheet, etc.
1. Compression in plane of sheets. Critical stresses.
pp. 40-41.
2. Shear in plane of sheets. Critical stresses, pp.
44-45.
3. Loads normal to plane of sheets. No known
data.
4. Combinations of above. No known data.
II. CURVED PLATES, RECTANGULAR
A. Isotropic.
1. Compression parallel to generating element.
Critical stresses, pp. 7- 9. See also Timoshenko,
"Theory of Elastic Stability," p. 467.
2. Shear, torsional or transverse. See N. A. C. A.
Technical Memorandum 774. Also Timoshenko, "The­ory
of Elastic Stability," pp. 48Q-490. Also N. A. C. A.
Technical Note No. 343.
3. Loads normal to generating element. See Timo­shenko,
"Theory of Elastic Stability," pp. 319, 445.
4. Combinations of above. See Timoshenko, "Theory
of Elastic Stability," pp. 475, 490.
B. Corrugated.
1. Compression parallel to the corrugations. Criti­cal
stresses, p. 40.
2. Shear, torsional or transverse. Critical stresses,
pp. 45-46.
3. Loads normal to generating element. No known
data.
2
4. Combinations of above. No known data.
C. Stiffened.
1. Compression parallel to stiffeners. Critical stresses
pp. 25- 26.
2. Shear, torsional or transverse. No known data.
3. Loads normal to generating elements. No known
data.
4. Combinations of above. No known data.
III. STIFFENER SECTIONS
1. Compression parallel to axis. Critical stresses,
secs. 4, 5, 6, 7, and 8.
2. Flexural loads. See standard texts on mechanics
for analysis of unsymmetrical sections. See Timo­shenko,
"Strength of Materials," vol. I, p. 192, for
analysis of open sections subject to twisting.
3. Shear loads. See standard texts on mechanics.
4. Torsional loads. See standard texts on mechanics
and "Theory of Elasticity."
5. Combinations of foregoing. See standard texts.
IV. MONOCOQUE SECTIONS, UNSTIFFENED
A. Circular sections.
1. Compression parallel to axis. N. A. C. A. Tech­nical
Report 473. Timoshenko, "Theory of Elastic
Stability", pp. 419, 439, 453. See also references in
N. A. C. A. Technical Memo. 785.
2. Bending. Critical stresses, p. 27. N. A. C. A.
Technical Note 479. Timoshenko, "Theory of Elastic
Stability," p. 463.
3. Torsion. N. A. C. A. Technical Note 427. N. A .
C. A. Technical Report 479.
4. Combinations of above. Compression-Torsion.
A. S. M. E. Trans., vol. 56 (1934), No. 11, pp. 795- 806.
Compression-Bending A. S. M. E. Trans., vol. 56 (1934),
No. 8, pp. 569-578.
B. Noncircular sections.
Fundamental data on noncircular sections are prac­tically
nonexistent.
V. MONOCOQUE SECTIONS, STIFFENED
A. Circular sections.
1. Compression parallel to axis. Timoshenko,
"Theory of Elastic Stability," p. 470.
2. Bending. Critical stresses, pp. 27- 34.
3. Torsion. See N. A. C. A. Technical Memo. 785,
p. 38.
4. Combinations of above. No known data.
B. Noncircular sections.
Fundamental data on noncircular sections are prac­tically
nonexistent.
VI. CONNECTIONS AND ATTACHMENTS
A. Rivets.
B. Welds, including spot-welds.
(Not considered in this progress report.)
VII. CORROSION PREVENTION
A. Ferrous materials.
B. Nonferrous materials.
(Not considered in this progress report.)
SECTION I. CRITICAL LOADS OF SMOOTH,
. UNSTIFFENED SHEET
The aeronautical structural engineer, in the design
of stressed skin or monocoque structures, must choose
between proportioning members carrying compression
or shear so they will not ,uinkle or so they will not
collapse until the maximum design load is reached. In
some places one criterion serves, and in others the other.
Not only must he proportion flat, curved or corrugated
sheets to carry the loads desired, but he must also pro­vide
stiffening members and other reinforcements to
furnish the necessary local stiffness required or to
distribute loads which are concentrated locally.
Occasionally the designer is seriously concerned by
the stresses causing local buckling or wrinkling of wing
or fuselage covering. He may wish to know whether
or not the buckling or wrinkling deformations of por­tions
of such structures under normal flying loads will
be sufficient to affect adversely the aerodynamic effi­ciency
and handling characteristics, the covering attach­ment
and adjacent structure, or the appearance. To
provide for the effects of "shear lag" or to determine
when the web of a spar ceases to be shear resistant and
starts to act as a tension field, the stress analyst must
know the intensity of stress at which buckles form. The
first part of the next section of this report deals, there­fore,
with the stresses which produce wrinkles in smooth
sheet.
These buckling stresses depend upon the method of
support of the edges of the sheet. A simply supported
edge is one that is constrained to remain straight
throughout its length, but is free to rotate about the
median line of the edge as an axis. Holding the edge
in a V- or Li-shaped groove or between round rods
appears to simulate this condition. A clamped edge is
one that is constrained to r emain straight throughout
its length without rotating. A sheet clamped between
heavy plates simulates this condition. A free edge,
as the name implies, is not restrained in any way.
SECTION 2. ISOTROPIC FLAT RECTANGULAR
PLATES COMPRESSIVE LOADS
Considering plates subjected to compressive loads,
we have those for which buckling is critical and those
which are to be stressed beyond the point where they
buckle to the absolute maximum they will carry. A
summary of formulas applicable to each is given below:
SECTION I. Conditions to produce b·uckling.
The general expression for critical compressive stress
at the start of buckling is:
where
K1r2E (t)2 Uc,. =12(1-μ2) b -- -- ----- ------1
uc,. = intensity of compressive stress (at the
start of buckling).
K = theoretical coefficient depending upon
sheet dimensions and type of edge
support (experimental data are in
reasonable accord) . See figures 1, 2,
3, and 4 for values of K.
E=modulus of elasticity of material.
μ= Poisson's ratio = 0.301 for steel and alu­minum
alloys.
t=thickness of sheet .
b= length of loaded edges.
a= length of unloaded edges.
In N. A. C. A. Report No. 382, "The Elastic Instabil­ity
of Members Having Sections Common in Aircraft
Construction," by Trayer and March, an expression is
given for the load producing buckling on sheets having
one free edge.
!tis:
P = [12(;~/L2i( ~)2+2(l~μiJEi where c is the
length of the half wave formed when the sheet
buckles. In most cases, c may be taken as the length
of the member, and the first term in the parenthesis
becomes negligible, so P=0.385 E i and the stress uc,·
= 0.385 E( ~)2- For small ratios of~ the value of le in
figure 4 may be taken as 0.456 and when substituted in
equation 1 gives u.,.= 0.412 E ( ~)
2
or about 7 percent
more than Trayer and March.
The critical loads or stresses given above are those a1.
which buckling due to compression starts. So far as is
known no method exists for determining the magnitude
of load which will produce stresses in the buckled sheet
approaching the yield point of the material and causing
the buckles to assume a permanent form. For many
purposes it would be desirable to design structures on
the basis of stresses just below those causing permanent
buckles to form, but this appears impossible at present.
For other purposes it is necessary to design to stress
intensities which cause failure of the member by crushing
or by its collapse due to column action. ,
SF.c. II. Conditions at maximum load, case I, two edges
loaded, all four simply supported. von Karman­Sechler
Method.
For flat sheets in compression the method described
by von Karman, Sechler, and Donnell in thefr paper of
June 1932 in the Transactions of the Applied Mechanics
Section, A. S. M. E., appears to be in best agreement
with test data. Briefly the method involves solving
equation 1 for the width b at which the critical stress
on the simply supported sheet is equal to the yield point
of the material. As first described K was taken as 4,
but later developments depend upon a variable K.
41r2E (t)2 Assuming K=4 we have, u.,=uw 12 (l - μZ) b '
and for aluminum alloy or steel μ=0.30, so
Uyp= 3.61EGJ
whence
b= l.90t' V/ E U yp
Tests made at the Bureau of Standards and recorded
in -. A. C. A. Report 3ji6, "Strength of Rectangular
Flat Plates under Edge Compression," by Schuman and
Back, showed that the load carried by flat sheets with
simply supported edges was essentially independent of
the width of the sheet, so von Karman and Sechler pro­posed
treating the sheet as though it had an area near
each edge which could be stressed to the yield point
where the central portion was practically unstressed.
4
' :l±
FIGURES 1-2.
5
',-
F IGURES 3-4.
6
FIGURE 5 .
The sum of these two stressed widths, called the
"effective width," was then taken as
b.=2W= l.90t.../E/<Tup
Tests made at M . I. T., C. I. T., and various other
places showed that 1. 90 was too high and indicated that
it was not a constant. A good average value for sheets of
normal size and thickness was found to be 1. 73 ; and
1.7 has been widely used with satisfactory results.
Substituting the empirical 1.7 for the more rational 1.9
and assuming that the sheet may act at a stress higher
than its yield point if it is attached to a stiffener or
otherwise supported, so that it reaches its maximum
load when the stiffener is carrying the maximum stress
of which it is capable, we have
b,= 1. 7t.../ E/<Tcr•
On the basis that the "effective" width, b., carries a
stress <Tcr, it is obvious that the load on the "effective"
area will be P = b,tr,,,, or P=l.7t2-,/Er,.,.
For normal sizes of sheet the above expressions for
"effective" width and total load on simply supported
panels are in satisfactory agreement with test results.
It has been found that stiffeners provide support, if
properly designed, equal to that assumed for the simply
supported edge condition.
For investigations requiring more refined computa­tion
it is desirable to vary the coefficient 1. 7 with
t: b-J EJ u ,,. The values established by Sechler are shown
in figure 5, the coefficient C1 being plotted against
t
A=b-J E/<Tcr•
The above expressions are summarized below for the
case of simply supported edges and for other edge con­ditions.
Considering cases I to IV as in section I:
CASE I, TWO SIDES LOADED, ALL FOUR SIMPLY SUPPORTED
Effective width, b,= 2W= Ct-JJ:l_
)' Ucr
Total load= P= Ct2.../ E""
C= l.90 by von Karman, theory.
P/2
] I
I
w..J
a~ I
I b ]
]!'1G"CTRE_9 ,
i%
......,
~
0
I ~ !.-w :::I I I lll
Q)
""'
~ -,.J
t-'l
P/2
C= 1.70 by Newell, empirical.
C= value from figure 5 by Sechler.
Cox, R and M 1553 and 1554, gives
= C't v[-;E;;.. + Db where C'=l.52 and D=0.09.
CASES II AND III. CLAMPED EDGES
Effective width, b.=2W=Ct'\/E
<Tcr
Tot al load, P= Ct2-,/ E""
C=2.51 on basis of K=6.96.
P/z
I
I
P/2
I I
w~ j+-w
____I - b ---+---
F IGURE 7.
b,= 2W
There are no test data to vindicate the above coefi­cients.
Cox in R and M 1553 and 1554, gives b.=2W
= C't V/ E +Db where C'=2.18 and D = 0.14. <Tcr
CASE IV,. TWO E DGES LOADED AND SIMPLY SUPPORTED,
THIRD SIMPLY SUPPORTED, FOURTH F REE
FIGURE 8.
Effective width, b.= Ct v/-E;;;;
Total load, P= Ct2../&.r
153696-40-2
7
C= 0.641, based on K = 0.456 at ( ~y =0
C=0.68 by von Karman.
C=0.60 by Lundquist.
P=[ 12<;~μ2) ~+2c1~ 0 J ~=[o.9( ~)2 +o.385]
Eifor μ= 0.3 and~ equal ratio of sheet width to length
of half-wave formed. Trayer and March, N. A. C. A.
Report 382.
Case I above has been checked against test data on
flat sheets of aluminum alloy and stainless steel by
several agencies and found to be in satisfactory accord
with the tests; so no effort will be made in this report to
present data for the purpose of justifying it. The
Trayer and March formula of case IV has been checked
for isotropic materials and modified for nonisotropic,
as recorded in N. A. C. A. Report 382. It appears to
give satisfactory results and should be used in prefer­ence
to the other formulas, since the others do not
provide for the reduction in load which occurs as b
increases. It is probable that the maximum load is
not entirely independent of the yield point of the mate­rial
as indicated by the Trayer and March formula,
but it is certian that it is not independent of the width.
SECTION 3. ISOTROPIC, CURVED RECTANGU-LAR
PLATES COMPRESSIVE LOADS
The problem of the curved plates in compression is
more difficult to treat mathematically, so there are few
rational formulas available for the determination of the
stress causing buckling or the stress at maximum load
on such members. Most of the formulas are modifica­tions
of those for complete cylinders, but Timoshenko in
"Theory of Elastic Stability," page 467, develops expres­sions
for the critical buckling stress on curved panels
and on page 470 states that the resulting equation,
"•r=0.6E~, is in satisfactory accord with test data on
aluminum alloy panels, provided the axial and circum­ferential
dimensions of the panel are about equal and
the central angle subtended by the panel is less than
about % radian. As the central angle increases to 2
radians the critical stress diminishes to about half that
given above and approaches-
.,: .._- cl.
FIGURE 9.
0.6 !._ 10- 7 ~
1
, R t
Uuzt = ~ - E
1 + 0.004 -
"""
8
as developed by Donnell for thin walled tubes in
compression.
Where
t=thickness of sheet.
R=radius of curvature.
o-uz,=ultimate stress in sheet.
o-cr= buckling stress.
u.,,=stress at yield point.
I_
~ I -
)'! . I b I
I
) I
I
' I
-
FIGURE 10.
-4
I
I
I
I
I
I
I --
Since airplane structures normally involve dimensions
placing some panels in one of the above categories,
some in the other and some between, there is little in the
above that is of general use to the designer.
SECHLER'S METHOD
The adaptation of the von Karman-Sechler formulas
for flat plates which was made for curved plates by
Sechler appears to be the best all round solution for
the stress at ultimate load. Reduced to its simplest
terms, Sechler's ~ .e,thod amounts to treating a curved
panel having simply supported edges as being composed
of effective widths, carrying the same stresses and loads
as in the case of a flat sheet, plus a central area which
works to a stress intensity of u= 0.3E{ The effective
width, as in the case of the flat sheet, is found from
b,=Ct fE where C is taken as 1.7 for average values
~~
or from figure 5 as the case may be. This width,
working to a stress, uc,, carries a load P' = Ct2-,/ Eu.,
while the area between the effective widths takes
P"=t(b-b,)( 0.3E~} The total load is then
P' + P" = Ct2
-,/ Euc,+ ( b-Ct~ u~)( 0.3E~) which may
be simplified to
P= Cct2-,/ Eu CT
where
Cc= C-0.3C1-1J+ 0.31J.
C= 1. 7 or coefficient from figure 5.
A= fEl , 7)= IE.!! '\/~b V Ucr R
The major drawback to this method is that no provi­sion
is made for panels of different lengths in computjng
the load carded by the central area of the sheet. Com­parisons
of predicted loads with results from tests on
c·urved sheet made at M. I. T. show the method to be
slightly conservative for panels 6 inches long, slightly
optimistic for 12-inch lengths and very optimistic, 25
to. 50 percent, on panels 18 inches long. 8jnce the load
carried by the sheet is normally small in comparison to
that in the stijfeners, the errors involved in computing
strengths of stiffened panels by this method are usually
less than 10 percent, but it should not be forgotten that
length has an effect on the strength of curved sheet.
As might therefore be expected, the method is in better
agreement with test results for sheets having large
radii than for those with small radii. It is also better
for wide sheets than narrow.
Table 1 shows the agreement between predicted and
test loads on curved panels as obtained by this method,
the data chosen being representative of that procured
on some 150 panels of 17 ST aluminum alloy tested at
M . I. T.
In considering the data in table 1 it should be no ;ed
that the test data are for most cases the result of a test
on a single panel of the dimensions shown and that
some of these panels had yield points higher or lower
than average, some were thinner or thicker than the
nominal dimensions, so it is necessary to make com­parisons
on the basis of trends rather than on specific
values. The agreement between predicted and test
values is good and the method simple to apply.
EMPIRICAL METHOD
Figures 11 and 12 (figs. showing factors K1, K 2, )
developed empirically from the 17 ST panels tested at
M. I. T. give coefficients by which the load computed
for a flat sheet may be modified to provide for the
effect of radius of curvature, length and width of
specimen. That is, the load on a curved sheet is
P.= JCK2Pr1at where the load on the flat sheet is
Pn., = Ct2-,/ Euc, as described above.
The use of figures 11 and 12 gives results whose aver­age
error is about the same as that obtained by Sechler's
method, but the range of error is greater. However,
for determining trial sections it will be found that loads
may be found by the use of these figures somewhat more
readily than by Sechler's method and with approxi­mately
the same results. The curves are included for
such purposes.
OTHER METHODS
Cox, British R and M 1553 and 1554, and Redshaw,
R and M 1565, give formulas for flat and curved sheet.
Redshaw's formula, as given in N. A. C. A. Technical
Memo 785, is
Uc,= 6(l~μ2) [ ~12(l - μ2J(riY +(~t)' +(;l)"J
It is seen that this expression makes no provision for
the effect of length. Applying it to one or two typical
panels shows it to be extremely optimistic. It gives
Pre- Test stresses
t R b dieted
stress L=6 L=12 L = 18
----------- - ----
0. 020 30 12 4,250 3,960 3, 340 2,580
.020 JO 12 12,650 7,950 7,710 6, 190
-H ~ I: [t ! ! H:HJ~
H+~~; ; i I t-l+ t-#1- t::1-t-t
H
I'
+-~ I
8± T h H: -w 11 ~ -+-++--H-++ r+ 4+H- , Htt,
I I
FIOURII: 12.
tt f-l I le-le
,.
I-
9
1~:I::- ,+
FIGURE 11.
TABLE 1
I Pre- Test loads ,
t b r dieted
load L-6'' L-12" L-18"
-------------------
0. 020 12 30 881 950 800 620
20 1,110 1,145 1,000 735
10 1,796 1,910 1,850 1,485
5 3,176 3,200 2,600 2,330
. 020 3 30 561 540 490 420
20 630 570 540 440
10 834 795 650 525
5 1,247 1,140 960 740
_ 033 12 30 2,360 2, 590 2,100 1,980
20 2,960 3,635 3,370 2,760
10 4,765 6,580 6,085 5,325
5 8,375 9,560 9,020 8,570
. 033 3 30 1,379 1,210 1,110 1,040
20 1,491 1,420 1,300 1,180
10 1,827 1,905 1,725 1,355
5 2,499 2,960 2,585 2,125
. 051 12 30 5,500 6,750 5,925 5,045
20 6,880 9,060 8, 000 6,815
10 11,000 14, 900 12,310 13,265
5 19, 240 19, 950 19,150 19,300
. 051 3 30 3,158 2,550 2,160 2,050
20 3,360 3, 135 2,765 2,255
10 3,960 5,000 3,820 2,900
5 5,160 5,065 5,010 4,190
SECTION 4. STIFFENER DESIGN
On the basis of the follqwing investigation it is
believed that stiffeners made from flat sheet may be
approximately proportioned by determining the crush­ing
stress for the cross section and, for members in the
short column range, substituting this crushing stress
for the yield point in the Johnson parabolic column
formu:la. The crushing stress should be checked by
t est in any case, but for approximate or preliminary
10
design of members the following analytical procedure
is suggested:
1. Assume a section which is composed of a series of
flat elements to be made up of fl.at sheets having
elastically restrained or free edges. Where the edge
of the sheet is bent 90° around a radius approximately
twice the thickness of the sheet, it may in most cases be
treated as the equivalent of a simply supported edge
which will carry a stress equal to the yield point of the
material. Edges having larger radii of curvature and
those adjacent to sheets which are wide in proportion
to their thickness buckle at stresses less than the yield
point and some corrective factor is necessary to pro­vide
for this effect. For two types of section sufficient
data were available to permit drawing an empirical
curve for the coefficient which gives good results. The
coefficient is actually a function of the torsional stiff­ness
of the adjacent sheets and edges, hence, should be
expected to vary with the radius and angle of bend, the
thickness and width of the material adjoining the edge
in question, the perfection of the edge formed, and such
other factors as may affect the elastic support furnished
by the adjacent edges and sides. As will be seen from
the tabulated data below, some shapes provide suffi­cient
support so that the effective widths near the
edges develop the full yield point of the material before
failing, others apP.ear to fail at about 75 percent and
some at a much lower fraction of the yield point.
It is believed, however, that for any given section a
form factor may be developed from tests on relatively
few specimens which will represent the conditions of
support for the various edges for varying widths of
sheet, so that the designer can predict the effect of
changes in cross-sectional dimensions with reasonable
accuracy.
2. Treat the elements, both edges of which are
bent, as simply-supported fl.at plates which will carry
a load of P = l.7t2../EX uc, if their center line width is
greaterthan the effective width found from b.= 1.7t -yfE~,
where u 0 , represents the yield point stress reduced by
the appropriate form factor to represent the relative
condition of edge support. When the center line width
is less than b., the load which the element will carry
will be P= b X tu0,, where b represents the center line
width.
For elements such as the outstanding legs of channels,
one of whose edges is simply supported, the other free,
theloadmaybefoundfrom P= [ 12(;"~ μz) -~+ 2(f + μ)J
E. f where s represents the width of the element, t its
thickness, c the length of half wave formed when it
buckles and μ= Poisson's ratio. For steel and the
aluminum alloys, μ=0.30 and c may be taken as the
length of the member when such length represents an
L/p for the member in the vicinity of 20. In most
cases, the first term is negligible and the load may be
t3
represented by P=0.385E 8 · For small values of s,
this expression indicates loads causing stresses beyond
u.,, in which case, the load should be taken as P=
aXtXuer.
An outstanding leg having a small flange or lip rolled
on it lies somewhere between a sheet having a simply­supported
edge and one having a free edge. As shown
by the tabulated data which follows, such a flange,
when its length is three or four times its thickness, is
usually sufficient to produce a simple support condition.
It is believed that the distance between rivets may be
used as the approximate length of half wave in com­puting
the load on the legs of stiffeners attached to
sheet, but the data available involve too many other
variables to permit establishing this as a fact.
When lightening holes are used, the strength of the
element in which they occur should be based on the net
area taken through the cut-out. If the holes are
flanged, the meager data available indicate that the
flange is approximately equivalent to a simply-sup­ported
edge and that the strength of the areas to each
side of the lightening hole may be determined on that
basis. Where the holes are not flanged, the strength of
the adjacent material appears to lie between that
expected of a free edge and that for a simply-supported
edge. The data available are too few to permit the
evaluation of the supporting effect in the form of an
empirical factor.
Bends of vee or other shape involving sharp edges
serve to break a wide, flat element into two narrower
elements having approximately simply supported edges.
the effectiveness of the "edges" varies, as would be
expected, with the size and shape of the groove, but a
reasonable approximation can be obtained by assuming
the groove to act as a simply-supported edge.
Curved elements, such as that in the ...IL-shaped
stiffener appear to develop the full yield point stress
for the material if their diameter/thickness ratio is 30
or less, and their point of tangency with the fl.at element
appears to act as a simply-supported edge for that
element.
Extruded sections having corner fillets have better
than simply-supported edges and it appears that the
crushing strength of their elements may be based on a
condition between simply-supported and clamped
edges. For many extruded sections, the crushing
stress will be equal to the yield point of the material.
3. Having determined the load which each element
of the section will carry, if acting alone under the as­sumed
conditions of edge support, the loads may be
summed up for the entire section and divided by the
area of the cross-section to obtain an average crushing
stress for the section. For stiffening members in the
short column range, the average crushing stress based
on an assumed length of member sufficient to give an
L/p of about 20 may be substituted for Fin Johnson's
. F2(L/p)Z
parabol!c column formula, Pl A= F - 4c1rz E , and the
stress determined for the design L/p. For members in
the long column range, the standard Euler formula
suffices.
It is assumed for the method outlined above that the
stiffeners will be attached to the structure in such a way
that they will not fail by combined torsion and bending.
Where such failures are possible, the likelihood of their
happening should be checked by the method given by
H. Wagner and W. Pritschner in N. A. C. A. Technical
Memo Nv. 784,1 "Torsion and Buckling of Open Sec­tions."
In making tests on stiffener sections to check
the predicted crushing stresses given by the above pro­cedure,
provision should be made to insure that the
failure obtained is a crushing failure and not a torsional
one. On members such as simple channels, this can be
done by preventing the rotation of the channel during
test by attaching a long stick to the specimen near its
midlength and applying a small correcting moment
when the section starts to twist. Such a device should,
of course, be attached so it will not alter the stresses on
the specimen and so it will not produce a supporting
effect of such character as to reduce the effective length
of the member.
SECTION 5. DETERMINATION OF CRUSHING
STRENGTH (STIFFENERS)
The theory of elastic stability may be applied to the
buckling of the various elements of a section or it may
be applied to the section as a whole. The method out­lined
below is based on the stability of the separate
elements and was proposed by Mr. A. B. Callender and
investigated in his thesis, "Elastic Instability of Dural­umin
Columns in Compression," M. I. T. 1933. It is
vindicated by Timoshenko in his "Theory of Elastic
Stability," page 333, where he shows the four sides of a
square tube to buckle as though each side were a com­pressed
rectangular plate with simply supported edges
and on pages 342 to 350 where methods of analysis are
developed for various conditions of elastic support of the
edges. Due to the complexity of the resulting equa­tions,
it is believed that designers will prefer the use of
empirical form factors and coefficients to the completely
rational method.
The method outlined by Messrs. W. S. Parr and W.
M. Beakley of the California Institute of Technology
in their paper, "An Investigation of Duralumin Channel
Section Struts in Compression," Journal of the Institute
of Aero. Sciences, volume 3, September 1935, pages
21-25, is based on the application of the stability equa­tion
to the section as a whole rather than to its ele­ments.
They establish three types of failure desig­nated
Euler, plate and torsional, and express the critical
stress as ffcr= KE(t/s) 2 where l( is a coefficient which
varies with the type of failure. The second condition,
or plate failure corresponds with the crushing strength
of the section as considered above and the value of l(
for that is :
2,,.2v( 1)2[cv+1)+4(l-μ)Vn
l( 12(1-1-'2)
where V = -J%1r2( 1)3 +3
s=length of outstanding leg of the channel.
b=width of back of the channel.
It will be noted that l( is a function of { only, so that
for channels whose ratios of leg to back are identical,
1 See also N. A. 0. A. Report No. 582, "Theory for Primary Failure
of Columns, Lundquist and Fligg."
11
the critical stress will vary as ( ~)2· A check of this
relation against the tests made by Roy A Miller and
recorded in A. C. I. C. No. 598, "Compressive Strength
of Duralumin CHannels," does not confirm this as fact,
but indicates the variation to be more nearly ( f )1'
than ( f) 2
• For the series of channels in which the
ratio f is 0.5 we have:
In terms of I Miller's Ratio CD' -J~
t Uu
b 8 8
-------
31,000 241 121 o. 0833 o. 00695 0. 288
28, 230 28t 14t • 0714 . 00505 .267
25,460 321 161 . 0625 . 00390 . 250
22,690 361 18t . 0555 • 00308 . 236
19,920 401 201 .0500 . 00250 .224
17,150 44t 22t . 0454 . 00206 . 212
14,580 481 241 . 0416 • 00174 . 204
13. 000 501 251 .0400 • 00160 .200
FIGURE 13.
As shown by figure 13 in which ffcr is plotted against
t ( t ) 2
s' S and ( St )t, the curve plotted for (t/s)}, I.S
practically a straight line where those for t/s and (t/s)2
most definitely are not. It must be concluded, there­fore,
at least in the light of these data, that the method
of obtaining crushing stresses from the properties of the
entire section instead of its elements is not in suffi•
12
ciently close accord with standard test data to be
accepted for design.
On the basis of these same tests the method of treating
each element as a separate entity, as was suggested by
Callender, gives the following results when applied in
terms of the system of dimensions shown in figure 14.
h
----------------------:i-i-
I s I t _L
FIGURE 14.
Using E = 9,730,000 and <Tup=37,000, the average
values given in A. C. I. C. No. 598 and considering a
channel having b= 24t, s= 12t and area approximately
48t2 we have:
b. for back=l.7../E/ <Tw t=28t. The widths of back
being less than b., P b= bX tX <T.,,= 24tX tX 37,000=
888,000t2
t3 t3
P.=0.385X EXb=0.385X9,730,000 X 12t=312,000t2
P,otai= Pb+2P,= 1,512,000t2•
Ptota l l,512,000t2
31 500 lb/. 2 <Tcru,h= Area= 48t2 ' in.
From tests <Tcru,h=31,000 lb./in.2
Table 2 shows this method applied to other sections
investigated in A. C. I. C. No. 598.
TABLE 2
Channel size b/s Predicted Test
P cru11hing P cr u1hi na
24tX12tXL - - - ---- - ---- ----- ------ - --- 2 31,500 31,000
2 28,350 28,230
2 23, 700 25,460
28tX14tXL - -------------- ---- ----- __ _
32tX16tXL _ ---------- ____ - - -- - - - - -- --
36tX18tXL _ ------ -- -- -- - - ____ -- -- - --- 2 20,400 22,690
40tX20tXL- --- --- - --- _____ ---- ____ --_ 2 17,800 19,920
2 15,800 17,150
2 14,320 14,380
2 13,480 13,000
44tX22tXL - -- - - - -- --- --_ - ------- --- --
48tX24tXL ___ ---- - -_ -- --- ----- - - --- __
50tX25tXL -- --- ---- ------ -_ ---------_
36tX12tXL - -- - -- --- ------- - ---- ---- -- 3 28,000 26,650
481X16tXL ___ -- ---- -- ________ ---- ---- 3 19,000 21,900
60tX20tXL _ --- -- --- --- _____ - - __ ------ 3 14,240 17,150
12tX12tXL _____ ---- ---- _______ __ __ --- 1 29,000 35,400
20tX20tXL ___ ---_ ---- __________ ---- __ 1 18,100 22,700
1 16,900 16,400
1 16,350 14,800
24tX24tXL _ ---_ ---- ___ -_ - -- - --_ ---- --
25tX25tXL ___ ------ __ ---- ------_ --- __
The agreement between Callender's method and
Miller's test results over rather a wide range of sections
is sufficiently good to justify further consideration of
the proposed procedure.
For some smaller channels of 17ST tested with "flat"
ends at M. I. T. the following results were obtained:
For the section:
b=0.715 in.; 0.75 in., over all.
s=0.484 in. ; 0.50 in., over all.
t=0.031 to 0.033 ; 0.032 average.
I= 0.00146 in.4
Area=0.053 in.2
p=0.16 in .
E= 10,350,000 lb./in.2 (average).
<T.,,=39,500 lb./in.2 (average).
TABLE 3
Load at failure
Length (inches) L/p
Predicted Test
1.00 6.25 1,780 1,740
1. 50 9.40 1,610 1,620
2.00 12. 50 1,550 1,570
3.00 18. 75 1,508 1,400
4. 00 25.00 1,494 1, ,Jso
6.00 37. 50 1,482 1,470
The "predicted" failing load was obtained from
Pback= <Tvpbt=39500X0.75 X 0.032=950 lb. , the actual
width of back being less than the effective.
P zca= [ 0.9(
0z)2 + 0.385 J 10,350,000 Xoi!
23
=
[
0
·_;;
5
+0.385 J 680
These channels were tested with flat ends and the
failures were all of the "plate" type, local buckling of the
back and legs, so they represented the crushing strength
of the section. As is obvious the agreement between
predicted and failing load is good although the "pre­dicted"
loads were figured from the over-all dimensions
of the members. It would seem more rational to have
used the center line dimensions as was done in the case
of Mi,Uer's channels. The agreement with test results
is still satisfactory if center line distances are used,
the "predicted" loads all being reduced and the method
becoming about 5 percent more conservative.
It is apparent from table 3 that the crushing strength
of sections having free edges varies considerably with
increasing length of section, the difference being about
15 percent in the above case for a range in L/p from
6.25 to 37.50.
The following data on the crushing strength of lipped
channels are based on tests made by the General
Aviation Corporation and studied by Walter H. Gale
at M. I. T. The pertinent dimensions, based on
figure 15, are given in the tables following.
(a) (b)
Figure 15.
The agreement between predicted and test results is,
in general, satisfactory for all specimens having lips
0.75 inch wide. It is fair for the specimens having
0.43-inch lips, being reasonably good for the thicker
walled specimens, but not so good for those with thin
walls. As the width of the lip decreases, the errors
increase and the agreements are poor for the 1 X 1 X 0, 188
specimens which have not only narrow lips but thin
walls as well. It appears, as would be expected, that
the less elastic support an edge has due to the thickness
and shape of the adjacent material, the lower is the
stress at which crushing occurs.
13
TABLE 4.-Figure 15a type
I
Crushing stress
b1 b, b, t L <Typ P, P, P, Total
load
Predicted Test
--------------- --------------------
3.23 1. 5 o. 75 0. 0795 6 38,600 6,800 14,600 12,~ 20, 600 38,300 37,200
3.23 1. 5 . 75 .050 4 37,000 2,625 2,625 9,180 26,600 26, 000
2.69 1. 5 • 75 .050 8 53, 000 3,150 3,150 640 10, 730 30,400 29,200
2. 69 1. 5 • 75 .050 8 39,500 2,720 2,720 640 9,440 27,600 25,500
2.69 1.5 . 75 .050 8 39,500 2,800 2,800 640 9,680 27,800 25,000
2. 69 1.5 . 75 . 050 8 11,700 1,495 1870 1440 4,115 12,050 12,600
2. 69 1. 5 . 75 .050 8 14,900 1,670 1 l, 120 l 560 5,030 14,300 11, 700
2. 69 1.5 . 75 .C81 4 38,600 7,075 14,700 12,350 21,175 36,000 35,000
2. 69 1. 5 . 75 .041 4 42,500 1,875 1, 875 350 6,325 23,500 21,800
2. 69 1.5 • 75 .049 3 46,000 2,820 2,820 605 9, 660 30,500 23,900
2. 375 1.25 . 406 .0513 6 36,000 2, 740 12,300 1 750 8,840 33,300 26,400
2. 375 1. 25 .430 .0208 5 38,500 465 465 81 1, 557 13,550 10,870
2.375 1. 25 .4375 .0529 6 37,400 3,235 2,475 865 9, 915 36,800 27,100
2.375 1. 313 .4375 .0613 5 31, 700 3, 650 2,550 1 850 10, 450 32,000 29,800
1. 53 1. 47 .375 .0625 5 37, 000 3,540 13, 400 870 12, 000 42,000 37,400
1. 375 1.0 . 3125 . 039 6 36,500 1, 600 11,425 1 445 4,895 35,600 32, 000
1. 375 1.0 .3125 . 025 6 40, 000 675 675 200 2,425 26,200 19,600
1.0 1.0 .188 . 0208 8 36, 000 450 450 1 141 1, 630 24,700 15,600
1.0 1. 0 .188 . 0241 4 36,000 607 607 1 163 2,150 28,000 21, 500
1.0 1. 0 . 188 . 0153 6 36, 000 245 245 73 880 17, 800 14,600
1 Load on area indicated is"" X b X t.
Assuming this to be a fact and considering the U
and '-n:_., sections tested by the Ford Motor Co., the
data for which are given on pages 22 and 23 of A. C.
I. C. No. 685, "An Investigation of Available Informa­tion
on the Strength Properties of Reinforced Skin
Construction," it is possible to develop curves showing
the effect on the critical stress of this variation in
elastic support. Figure 16, for instance, presents
curves obtained empirically, which show the variation
in the critical stress for different ratios of width to
thickness of the back of the channel. They are in the
nature of form factor curves applying to the particular
shapes and sizes covered by Ford's U and '-n...:.i sections.
One interesting thing which they show is that the
variation in the form factor for each type is roughly
linear so that each could have been established from
tests on two or three sections and the strengths of
channels of intermediate gages or dimensions could
have been predicted with sufficient accuracy for pre­liminary
estimates and design without the need for
an elaborate series of tests. Where adequate data are
available, more accurate results become possible. While
the development of such curves for each shape of
stiffener proposed is admittedly a task, it is not a
difficult one and it is believed that enough can be
learned by comparing the curves for several typical
stiffeners to show which features are advantageous and
which deleterious.
It would appear from figure 16 that it is an advantage
to turn the flanges of channels outward instead of
inward since the critical stress will not drop off so
rapidly with increasing ratios of back width to thick­ness,
nor will it drop so low in the range of normal sizes
of section. Applying the coefficients of figure 16 to
the sections, Ford R and U series of sections, the
dimensions for which are indicated in figure 17, gives
the predicted load values shown in table 5 which,
when compared with the test loads, are seen to be in
reasonable accord. All are based on E= 10,000,000
.o.,,=40,000 lb. /in.2
TABLE 5
I
Crushing load
Section b1 b2 b, b, t P1 P , P, P1
--u----1-~-~--~1--;;--;;-~-- Pr:~:::ed ::lt5
U 1 I }4 • 028 754 754 208 2,678 2,820
U 1 1 }4 ------ - .035 to 0.065 work to " "=40,000 lb.fin.•
u 2 2 716 ------- • 035 568 I 568 I 60 1------- 1,860 1. 680
U 2 2 710 -- ----- . 049 All elements work to ""=14,000, test stress=l3,500.
U 2 2 H ___ ____ .065 All elements work to ""=20,000, test stress=20,000.
U 2 2 }4 ---- - -- • 083 All elements work to ""=28,000, test stress=28,000.
u 2 2 % ·------ .095 All elements work to ""= 40,000 lb./in. 2 I u 2% 2 !Vi• _______ .049 1,410
1
1.155 I 96
1
_______
1
3, 912 3,670
U 2% 2 H - ------ . 065 3,040 2,260 255 ------- 8,070 7, 450
U 2% 2 }4 --- - -- - • 083 5. 960 4, 130 450 --·· - - -- 15,120 14, 300
U 2% 2 % __ __ ___ . 095 All elements work to ""=34,000, test stress=33,500.
RR 11 22 %% 771106 .. 002305 I, 306800 I l, 316705 I 376000 1110900 I 52,,020300 I 15,,935800
R 2 2 % 716 . 035 1,120 l, 120 525 143 4,690 4,600
R 2 2 % 71 o . 049 2, 185 2, 185 88.5 230 8, 785 8,800
R 2 2 ')4 H . 065 All elements work to ""= 32,000, test stress=32,500.
R 2% 2 % H . 049 1, 000 2, 000 1, 030 265 8, 580 8, 650
R 2% 2 ~• M . 065 3,810 3,520 1,590 400 14,830 14,550
U 1 1 ')fo - - ----- .0208 320 320 70 ----- - - 1,100 1,030
Uu 1 1 ?io --- ---- . 0241 510 510 115 -- --- -- 1, 750 1. 640 1 1 Mo __ ___ __ .0153 160 mo 45 _____ __ 570 120
14
FIGURE 16,
The agreement between predicted and test loads in
table 5 is excellent as might be expected because the ·
form factor curves were developed to fit the above
t ests. However, the data do show what can be done
with the proposed method for predicting crushing
stresses when a few tests on short lengths of specimens
of the desired shape are available.
Returning to a consideration of the channels having
flanged lightening holes, figure 15b, and treating them
as having no form factor because of their properties,
the following results are obtained:
The net area of the back is assumed to carry a stress
equal to the yield point of the material because the
widths between the flanged hole and the outstanding
legs are less than the "effective" widths for the thickness
of material used. This appears to be a reasonable
approximation and is the equivalent, so far as crushing
goes, of assuming a flanged slot running the length of
the channel instead of a series of flanged holes; hence,
we conclude that the area between lightening holes
carries little compression, but serves primarily to make
the two sides of the section act as a unit. The outstand­ing
legs are treated as flat plates with simply-supported
edges and the lips as having one supported edge, one
free. See figure 15.
u R
]<'JGUHE 17.
The proposed method of analysis has been applied to
sections having shapes as shown in figure 18 with errors
of 10 percent or less. The stiffener shown at A was
analyzed as three flat plates having simply supported
edges plus two plates having one edge simply supported,
one free. Due to the flanged edges on stiffener B, it was
treated as five pl3.tes having simply supported edges
while C, for the same reason, was treated as three.
There are not enough data to permit the determination
of the length of flange required to provide the equivalent
of a simply supported edge, but it would appear to be
small.
lJ 1Lr
l fl
X , F r
FIGURE 18.
T ABLE 6
Crushing stress
b, b, b, F t P, P2 P, Total
"» load
Predicted '.l.'est ---- - --- - - - ------ - --------- ---
3.23 I. 5 0. 75 2. 5 0.080 38,600 2,240 4,600 2,300 16,040 38,600 { 45,000
44,000
3. 23 1.5 • 75 2.5 . 0465 37,000 1,720 2,625 640 8,200 21,400 { 32,700
29,600
2. 69 I. 5 . 75 I. 31 .0408 42,500 1,875 1, 875 350 6,325 28,400 27,300
2.69 I. 5 . 75 1. 31 .080 38,600 4,950 4,625 2,320 18,840 38,400 { 40,600
35,000
The agreement between predicted and test results is good with tbe eicception of tbe first specimen for wbicb tbe yield point stress given in the
test data is undoubtedly in error.
Stiffeners of type D were analyzed on the basis of
being a half tube plus two angles but the results were
unsatisfactory. They were then regarded as a curved
element which would carry a stress equal to the yield
point of the material for normal ratios of diameter to
thickness, two flat sheets having simply supported
edges and two having one edge simply supported, one
free. It may not appear reasonable to assume the edge
of tangency between curve and flat as the equivalent of
simple support, but the agreement between predicted
crushing stress and test data was good, a representative
case studied by Sousa and Greenwood in their thesis,
M. I. T ., 1934, having shown an error of 6 percent.
Aluminum alloy stiffeners similar to A were attached
to sheet as shown by F in figure 18, and tested by the
Boeing Aircraft Co. For sections having the dimensions
shown in figure 19, table 7 gives a comparison between
predicted and test loads. In obtaining the "predicted"
values, the outstanding leg of the stiffener was treated
as a sheet having one edge simply supported, one free,
even though it was attached to the skin because the
rivet pitch used, from 1 to 1.5 inches, appears to have
been sufficient to permit buckling of that edge between
rivets. The skin was treated as a simply · supported
sheet and, due to its being attached by two rows of
rivets, was assumed to carry a stress equal to ucr for
the section if its width were less than the effective
width. Since the test report gave no values for E or the
yield point, E was assumed to be 10,000,000 and the
yield point was taken as 36,000. The latter value was
modified by a form factor of 75 percent and <rcr taken at
27;000. This gave good results in three cases: Fair in
one and poor in one, the unfavorable results occurring
with the thin-walled sections, or with those having a
high b/t. A greater reduction in <rcr should apparently
have been made in both cases to provide for the low
edge supporting effect of the elements of these sections·
FIGURE 19.
Stainless steel stiffeners of type E were treated as a
series of flat plate elements but with a greatly reduced
TABLE 7.
Crushing loads
b, b, b, b, t, t,
Pre- dieted Tests
------------
l),;\ % % % 0.020 0.025 1,580 1,730
l~i, % % ')ii .020 . U20 1,380 1,140, 1,075
l ),;\ l ~32 l'l6 Y.12 .020 .U2!i 1,720 1,700, 1,727
I ~i, 1732 yi6 1~2 .U25 . 020 1,940 1,860, 1,830
1)16 ' %2 H, % .025 . 020 ~.375 1,690, 1,740
153696-40-3
15
value of E, 16,000,000 having been found to give good
results for the cases tried. This was, of course, the
equivalent of using a form factor to provide for the
elastic conditions at the edges and in the grooves, but
too few specimens were available to permit constructing
a set of curves similar to those of figure 16. However,
representative results show predicted loads of 13,000,
12,450, 15,100, and 14,000 compared with crushing loads
of 12,485, 12,080, 14,320, and 14,010 pounds, respec­tively,
obtained from tests.
Data obtained from one manufacturer on square
17ST tubes having E=I0,000,000 u.,,=46,000 lb./in.2
follow. The tubes were analyzed as four simply sup­ported
fiat plates.
TABLE 8
t Width Test,• P Predicted,
of side, b p
Inches
0.040 4 8,000 7,400
.040 3 7,750 7,400
. 041 2 7,600 7,780
.040 1 7,300 7,360
1 Each value is the average of 3 tests.
b.= 1.7 r.E t= 25.5t= 1.02" for t=0.040, 1.045" for V <Tvv
t=0.041.
P for each side=I.7.,/Eu vv t2=1,850 for t=0.040, 1935
for t=0.041
Since the one-inch sides are less than the effective
width, P=1XtX46,000 or 1,840 pounds for each,
giving the predicted P for the one-inch square tube as
7,360.
Similar tests made at M. I. T. on small square tube,
%6-, 1~6-, 1%6-, and 1 ~ 6-inch sides, gave higher crushing
stresses in most cases where the width of side, b, was
less than b,, than the yield point and in some cases
higher than the tensile strength of the material aa
determined from strips cut from the sides of the tube.
For such sections it would appear conservative to
compute the crushing stress as the product of the area
times the yield point of the material in the sides. It is
believed that the work done on the material in forming
the corners of the tubes increases its strength properties
considerably so that the corners actually withstand
crushing stresses equivalent to the tensile strength of
the flat faces. Tests made by Boeing and recorded in
reports of tests Nos. 14575 and 13889 show similar
effects on tubes whose sides were less than b. in width.
For the wider tubes, in terms of b/t, we have from
the M. I. T. tests on 17ST tubes having E=107 lb./in.2
TABLE 9
b.=
Pre·
Width Wall Load per side dieted• Test
of tuhe, thick· L/p 1.7-J El 1.7 1.>.,/E<1,, crush- crushing
b ness, t "" ing stress
stress
----- ---
o. 9375 0.031 21. 7 0. 79 1,075 35,000 42,250
. 8125 .016 18. 6 .40 288 22,000 22,200
.6875 .016 21.8 .40 288 25,300 24,000
. 5625 .016 17. 9 . 40 288 32,000 32,200
1 "'•• based on a_veraged value obtained from tests of strips from sides of
tubes=44,000 lb./m.2
16
For rectangular tubes of 24SRT aluminum alloy
Boeing test report 14575 gives, with E-10.5X 106
lb./in.2
TABLE 10
Load Load Based Crushing load
Outside Wall dimensions thick- b, on wide on oar- on of tube ness, t sides, row yield each sides point dPiertee-d Test
--------------- --
2"1ox'Yi• 0. 0495 22. 5t 3,300 2,260 160,000 11, 120 11,800
21Yiox2~2 .098 21. 6t 13,300 13,000 165,000 52,600 53,100
1 "'" based on tests of strips from sides of similar tubes varying between
60,000 and 66,500 lb.fin.'
For "barrel" sections of 24SRT, the same Boeing
report gives-
T ABLE 11
Load Crushing load
Outside Wall Load on
dimensions thick- b, on flat curved Y. P.
ness sides sides Pre- T est dieted
------------------
2H• X 2•%2 •.. o. 050 21. 7t 3,490 4, 815 6'1, 300 16, 610 21,400
21Yio X 12%2 •. . 0585 21. 6t 4,800 6,750 65,000 23,100 25,500
21Yio X 2~2 ... .0565 23. Jt 4,200 5,950 57,000 20,300 25,600
21Yi• X 22'),;2 .. . 065 23. 21 5,530 6,470 56,500 24,000 33,200
The loads on the flat sides of the barrel tubes were
computed as though they were flat sheets, P=l.7
t2-,/ Eu uP, and the load on the curved sides was taken
as the area times the yield point, it having been assumed
that the curvature was sufficient to cause the material
to work to the yield point of the material. The pre­dicted
results are low by approximately 25 percent.
However, if the entire tube area be multiplied by the
yield point of the material, a good agreement is obtained
between predicted and test values. Hence, it may be
that by cambering opposite sides of a rectangular
tube, the stability of the entire section is so altered that
crushing does not occur until the average stress on t he
cross section reaches the yield point or it may again be
that the strength of the corners exceeds that of the
faces so that t he average strength of the complete tube
exceeds the value expected from the properties obtained
on strips from the sides.
SECTION 6. THE STABILITY OF OPEN SECTIONS
As the stress on a column approaches its ultimate
strength the cross section usually undergoes distor­tions
which may be local or may extend over the length
of the member. Such distortion of the section causes a
shift of the axis of resistance of the member from the
axis of loading so moments are created by the eccentric­ity
of the axial load and these moments cause the
member to deflect with the result that a shear is devel­oped
on the affected sections whose magnitude is P
sin i where i represents the slope of the bent section, P
the axial load on the column. Such shear stresses
produce forces on the elements of an open section which
may cause it to rotate as a whole or which may aggra­vate
the bending of the deflected elements. Where a
stiffener is fastened to a sheet these forces are normally
insufficient to cause the combination of sheet and stiff-ener
to rotate about the center of twist of the stiffener
but they may suffice to bend the elements of the stiffener
itself and cause it to fail in a combination of bending
and twisting.
F igure 20 shows representative shapes of stiffeners
and the direction of the shear forces acting on their
elements when the axial load is applied eccentrically.
A study of figure 20 shows why sections such as
channels sometimes fail by the legs bowing inward or
outward if they do not fail by twisting as a whole.
When bending subjects one leg to compression; the
other to tension, the section tends to twist as a whole as
shown at (a). When the eccentricity is along the other
axis the tendency is to fail by bending the outstanding
portions inward or outward due to the shear forces, as
shown in (b) to (f), the direction depending on whether
the point of loading is one side or the other of the
centroid and on whether such flanges as may be used,
bend inward or outward. Sketches (e) and (f) do not
present a true picture of the shear forces acting since
they are unsymmetrical sections. They would tend
to deflect about both of the axes shown due to a moment
in the plane of either, the axes not being principal axes
for these sections.
Cen~ofload
I <.. >
TendenC'f fa twist
Centroid~ ;::.oinf of lo.>.d
(¥)
I (C)
re.ndenC'f same i1. S
0b'. Greater
bending is part iil.ll'f
compensa.ted b'f 9u•fer
stillness of flilnge.
c'entroid~Point of load
(e) ~
Cent~id+o~f of load
I l
I (bJ
rendenc'f fo bend about center
of back and cause le9s to
cotl .. pse in ward.
:t~L - I (dJ -
l1oment .>.bout center of back
reduced b'f forcts on f/an9eo
a.nd stiffness of outstanding
legs is increased b'f the
flanges.
c'enfroi'd~Point of load
(f) ~
rendency lo rot..te about Tendency to twist.
pla.ne"c"
FIGURE 20.
It is seen, therefore, that open section stiffeners may
fail due to secondary moments and shears induced in
them by eccentricity of the loading or by bending of
the stiffener and the material to which it is attached.
Due to the fact that the least radius of gyration occurs
about one of the principal axes rather than the axis
parallel to the sheet some stiffeners tend to fail as
columns by buckling about a principal axis. The skin
reduces this tendency so the effective L/p is greater
than the minimum, but the exact effect has not been
evaluated. It should also be noted that tests of certain
sheet-stiffener combinations made on flat panels will
not indicate the true strength of such combinations
in a wing or fuselage where bending of the structure
produces a curvature in the panel which with unsym­metrical
sections causes them to rotate and prod~ce
secondary bending and torsional effects, causing them
to fail at stresses which may be materially below those
obtained on the flat panel. In developing shapes of
stiffeners this feature should be considered since it is
impossible to predict the magnitudes of the secondary
shears and moments developed by such accidental
but unavoidable eccentricities; hence it is impossible
to determine the stresses produced in the stiffener by
them. Effort should be made to avoid unstable shapes
where possible.
SECTION 7. COLUMN STRENGTH (STIFFENERS)
The foregoing procedure for computing crushing
stresses, while not conclusive because of the limited
test data against which to check it, gives a close enough
approximation to the crushing strength for channels,
square and rectangular tubes to warrant further con­sideration,
especially since it has been found that for
many shapes a good approximation to the column
strength can be obtained by use of the Johnson-Euler
curves, the yield point as normally used in the Johnson
formula being replaced by the crushing stress for the
section. When making such use of the column formulas
it is necessary to select a reasonable value for the fixity
coefficient to provide for knife-edge or pin-end condi­tions,
"flat" ends, and similar conditions.
17
t.
Figure 21 shows the results of a series of tests on
channels of the same size as those considered in table
3. The points indicated by circles represent tests made
between knife edges to insure a fixity coefficient of 1.0.
The data from table 3 have been added as crosses and
it is apparent that for the very short lengths there is
no difference between the channels having "flat" ends
and those having pinned-end conditions. The differ­ence
begins to show up at an L/p of 25 or 30. At an
L/p of 20 or thereabout the failing stress is so near
the crushing stress for the section that a Johnson pa-b
1 f l f
(<Tcr)2(L/ p)2 h .
ra O a O t1e orm <Tc = <Tcr - 4c1r2E 'were <Tc JS the
stress the member will carry as a column and <Tcr its
crushing stress, is in very good accord with the test
data. The members of high L/p lie near the Euler
curve as would be expected. Figures 22, 23, 24, 25,
will facilitate the use of these formulas.
It is believed, therefore, that a reasonable approxi­mation
to the strength of open section or other shapes
of stiffeners may be obtained by determining the crush­ing
strength of a section whose L/ p is about 20 by
test or by the analytical method suggested above, and
by using this crushing stress in the Johnson column
formula for obtaining the allowable stress for other
slenderness ratios. T t is further believed that the stress
so computed represents the critical conditions for
stiffeners attached to sheet and that the condition of
torsion and buckling contemplated by Wagner and
Pretschner in N. A. C. A. Technical Memo 784 need
FIGURE 21.
18
FIGURE 22. FIGURE 23.
FIGURE 24. FIGURE23;
19
not be considered with normal stiffeners normal1y
attached.
To verify the procedure proposed for determining
column strengths from crushing stresses the following
tables are offered:
Considering square tubes first, taking those from
Boeing test report 13889, whose walls are so thick in
comparison to their widths that the crushing stress is
equal to the yield point of the material.
TABLE 12
Failing column
Crush- stress
Tube size Material ing L/p C
stress Predict-ed
Test
- - - ------
2:){s x 2:){o x 0.094 ___ H. T . s teel. 151,000 53.6 1 94,000 84,000
2:)1' 6 X 2:){s X 0.097 ... ___ __ do. ____ 160, 000 39. 1 1 128,000 117, 000
2:)1' 6 X 2:){o X 0,140 ... _____ do _____ 181,000 37.0 1 141, 000 140,000
2:){o x 2'Vio x 0.142 ___ ____ _ do .... . 182, 000 54. 7 1 97,500 113, 000
21),is X 21 ),io X 0.240. l7SRT _____ 75,500 54. 5 1 33, 000 32,000
2'),is X 21 ),io X 0.240. 17SRT ..... 75,500 25.0 1 66,000 63,000 •
Taking the 24SRT rectangular and "barrel" sec­tions
of Boeing report No. 14575 whose crushing
strengths were considered in tables 10, 11, " "e obtain:
T ABLE 13
Crush-
Tube size Shape ing L/p
stress
--
2')1o X lJ,io X 0.0495 ... Rectangular. 1 40, 500 24. 7
2:){o X lj,15 X 0.0495 ... _____ do ______ 142,300 31. 5
2')1o X lj,15 X 0.0495 ... ..•.• do ______ l 41,300 45. 2
21),is X 2~2 X 0.098 ... _____ do ______ 162, 600 20.6
21),is X 2~2 X 0.098 ... _____ do. __ ___ 162, 600 33.3
21 J.i 6 X 2~ 2 X 0.098 ... ___ __ do __ __ __ 162,600 46.0
2lJ,io X 12%2 X 0.0585_ Barrel.. _____ 262,400 20.4
WJ.io x 12%2 x 0.0585-1 .... . do ______ '62, 400 30.4
' t.1 )-io X 12%2 X 0.0585-
1
..... dO ...... '62,400 40.5
21!,is X 2~2 X 0.058.L . ... . do ..... . '59,000 20.8
2'J.io X 2~2 X 0.0585 . ...... do ...... '59,000 29.0
21!,io X 2~2 X 0.0585 ....... do ____ __ '59,000 45. 5
' Based on predicted loads from tables 10 and 11.
' Test ,·alue.
Failing column
stress
C
Predict-ed
T est
- -----
1 37,000 39,200
l 37,500 35,800
l 32,000 29,900
l 58,000 65,600
l 51,500 53,400
1 42, 000 42, 300
1 58,000 63. 300
l 53,500 56,600
1 46, 250 48,900
l 55,000 57,600
1 51,000 52,300
l 40,500 41,200
For the channel sections of A. C. I. C. 598 Miller
recommended the substitution of the crushing stress
for the yield point in the Johnson column formula as
has been suggested above, since that gave the best
agreement with test data.
Two other series of channels of l 7ST alloy whose
back width is recorded as b, leg width as s, and thick­ness
as t, gave the results shown in table 15. No fo rm
factor was used with either series.
For the lipped channels as tested by ·General A via­t
ion we have results as tabulated below:
For this series of tests the short and long members
are in good agreement, but there are large discrepancies
in t he middle range. The data for yield points were
not obtained for each specimen but were average values,
hence may not have been correct for some of the speci­mens.
The predicted crushing stress from which the
column stress was determined was based on the full
value of the yield point whereas, as has been noted
above, a better approximation for some sections has
TABLE 14.-For channels of fig. 15a
Predict- Stress of ulti-ed
mate load
b, b, b, t L/p crush-ing
stress Predict- cd Test
----------------- ---
3. 23 1. 5 o. 75 0. 0795 6 38,300 38, 100 { 36,200
39,100
3.23 I. 5 . 75 . 0795 10. 25 38,300 37,900 { 33,500
34,100
3. 23 I. 5 . 75 .0795 16 38,300 37,350 { 31,200
29,650
3. 23 1.5 . 75 .0795 25 38,300 36,000 {29,600
29, 850
3. 23 1.5 . 75 .080 34 38,600 34. 300 I{ ~: ~
3. 23 I. 5 . 75 .0795 58 38,300 25, 800 23, 900
3. 23 1.5 . 75 . 0795 82.3 38,300 13, 100 13, 700
been obtained by using a reduced yield point to provide
for the elasticity of the edge supports. Had the
crushing stress been based on a yield point modified by
a suitable form factor, the agreement with column
tests would have been improved .
T ABLE 15
Pre- Stress at ulti-dieted
mate load
Length b s t criunsgh - L/p
stress Pre- Test dieted
-------- - -----------
2. 80 0.65 o. 70 o. 053 31,300 13. 3 30,850 31, 600
4. 00 . 70 . 72 .052 29,600 17. 9 28,900 30, 800
6. 30 . 72 . 73 .052 28,800 27. 5 27,100 28,490
8.00 . 70 . 72 . 052 29,610 35. 8 26,750 27, 530
10. 62 . 72 . 75 . 052 28,600 45. 7 24,200 23,720
12. 73 . 70 . 72 . 052 29,610 57.0 21,950 21,730
14. 62 . 70 . 72 . 052 29,610 65. 4 20,000 17,930
4.08 1. 75 .84 . 053 24,000 15. 5 23,750 28,300
6.90 1. 72 .88 .053 22,600 24.5 21, 750 24, 480
9. 60 1. 75 . 93 .053 22,200 32. 5 21,000 23,250
15.15 I. 72 . 97 . 052 20,600 47. 2 18,000 21,960
17. 95 1. 76 . 92 .052 21,450 60. 7 17,180 17,510
20.68 1. 75 .91 . 052 22,000 70. 2 15,850 16, 480
23. 45 I. 74 . 92 .051 20,000 80.2 13,250 14, 170
For the channels of figure 15b, we have:
T ABLE 16
Pre- Stress at maYi-dieted
mum laod
b' b 2 b 3 F t L/p crush-ing
Pre- T est stress dieted
--- - ------- ---
3. 23 I. 5 0. 75 2. 5 0. 05 6 31,400 31,300 1 3229,, 670000
3. 23 1.5 . 75 2. 5 .05 10 31,400 31,140 30, 400
29,300
3.23 I. 5 . 75 2. 5 .05 16 31,400 30,760 { 30, 150
28,500
3. 23 I. 5 . 75 2. 5 .05 25 31,400 29,840 { 28,300
29,700
3. 23 I. 5 . 75 2. 5 .05 34 31,400 28,520 { 24,900
25,300
The above table shows a reasonable agreement
between predicted and test results though t he crushing
stress for the section was based on the full yield point.
The stainless steel stiffeners, type E of figure 18,
when checked for other values of L/p than those used
for determining the crushing stress gave the following
results as recorded in the thesis, "A Design Procedure
20
for Thin Stainless Steel Sheets in Compression," Stark
and Seary, M. I. T. 1935.
TABLE 17
Thick- lVfaximum loads
Length ness of L/p Fixity co-sheet
efficient Predicted Test
--- ------ --- -------
3.48 0.019 15.6 13,000 12,485 1
4.49 .019 20.1 12,450 12,080 1
7.54 .021 33.fi 10,450 10,240 1
10.70 .019 47.9 7,300 7.500 1
15.32 .021 68 .1 4.750 4,630 1
19.97 .019 89.4 3;000 2,765 1
.97 .021 4.3 15,100 14,320 2
1.46 .021 6.fi 14,900 13,800 2
6.00 .021 26.7 13,200 13,400 2
9.23 .019 41.3 11,150 10,860 2
13.81 .019 61.5 9,750 9,040 2
18.47 .019 82.6 6,900 8.480 2
The critical stress was taken as 145,500 lb./in.i in pre­dicting
the above loads. The first series was tested with
pin ends giving a fixity coefficient of one and the second
series with flat ends. The agreement between predicted
and test loads is good with the fixity coefficient assumed
to be two.
SECTION 8. NOTES ON STIFFENER DESIGN
From the foregoing discussion it appears that the
crushing strength of a formed section may be predicted
with fair accuracy by considering the stability of each
of the elements of the section having due regard for the
conditions of elastic support at the edges of each such
element contributed by the adjacent sides and edges.
In the present state of the art it seems desirable to use
empirical coefficients in the way of "form factors" to
account for the varying degrees of support obtained
rather than deal with the complex equations provided
by theory. In the future it may be possible to develop
charts empirical or theoretical which will evaluate the
contribution of various shapes of section to the stabil­ity
of each edge so that critical crushing stresses may be
predicted accurately. In any case, it seems desirable
to use as sharp corners as possible without damaging
the material and to keep the ratio of width to thickness
of each element as low as possible so that it will stabil­ize
adjacent elements and withstand high stresses before
failure.
The latter condition, however, is opposed to the re­quirements
established for sections which fail due to
column action rather than crushing. For such sections
it is desirable that the material be spread out so the
radius of gyration is as great as possible. A compro­mise
must therefore be effected between the require­ments
that both crushing stress and radius of gyration
have their maximum values. For thin walled, or lightly
loaded shapes this may require the use of grooves to
stiffen wide, thin areas or lightening holes to reduce
their weight. Where grooves are used it appears desir­able
to have them sharp cornered so that they will pro­vide
a stiff edge and cause the element in which they
are made to work to a high stress intensity. Where
lightening holes are used, it seems advisable to flange
them and to have them of sufficiently small diameter
that they do not cut out any of the effective width of
the material. Of the two methods for increasing
strength or reducing weight, it is believed that the use
of grooves with thin sheets is more satisfactory than
the use of a heavier sheet and lightening holes. This
item would provide a field for research, but does not
appear to be of sufficient importance to demand imme­diate
action.
The use of curved elements instead of all flat ones
appears to stabilize some shapes of stiffeners and cause
them to develop high crushing stresses. 1 t is probable
that they restrict the buckling of the flat sheet to one
direction, that is, it must buckle so that its tendency
is to reduce the radius of the contiguous curved sheet
but not to increase it, hence cause it to develop higher
stress intensities before failure.
Closed stiffener sections offer rlistinct advantages
over open sections because of their increased stability
at high loads or when they are bent due to the distortion
of the sheet or structure to which they are attached. It
is appreciated that ·such sections are liable to corrosion
from the inside out and that they are difficult to attach,
but from the standpoint of strength they are superior
to the open section which, when once deflected, develops
internal shear or compressive forces tending to twist it
or to deflect it still further.
Designers should consider these effects in determining
the allowable stresses to be used with such sections, since
tests made on the stiffener alone may give appreciably
lower strength properties due to the instability of the
elements of the section than will be developed when the
stiffener is attached to the structure, assuming, of
course, that the method of attachment tends to stabi ­lize
the critical elements. On the other hand, stiffener
sections which are unsymmetrical may be expected to
fail at lower stresses in an actual structure than in a
test. As normally used they are constrained by the
material to which they are attached to bend in a plane
which does not contain one of the principal axes of t he
section so they are forced to bend in a direction normal
to that plane as well. Such stiffeners roll over readily
and may not develop as high loads when deformed as
part of a structure as when tested on a panel under a
direct compressive load.
SECTION 9. STIFFENED FLAT PLATES IN
COMPRESSION
EMPIRICAL METHODS
Approximate Method.
Mr. H. W. Gall, in 1930, found from tests of stiffened
aluminum alloy panels that he could make a very good
approximation to the load obtained in test by assuming
the stiffeners acted to break the panel up into a series of
simply supported sheets, each of which carried a load
equal to that found by Schuman and Back in N. A. C. A.
Report No. 356. By adding the load carried by the
stiffeners to that carried by the sheets, assuming the
two to act independently, a good agreement was ob­tained
between predicted and test values.
For preliminary design a modification of Gall's
original method which requires a minimum of computa­tion
will be found very useful in obtaining approximate
sizes of members. T o the load on a simply supported
sheet, found from P= Cte-JJ§;;-;;, add the column load
for the given length stiffener on the basis of C= 1 for
pin-end condition, C= 2 for fiat ends, u er being the
stress at which the stiffener would fail if tested as a
simple column, that is, without the sheet. The sum of
these two loads divided by the area of sheet and stiffener
will give an average stress at maximum load for the
combination which is in close accord with test data.
The strength of the stiffener may be obtained by the
analytical method described in articles 4 to 8 of this
report or from a column test. The following example
will serve to illustrate the procedure.
F IGURE 26.
I0 = 0.00146
A = 0.053
p = 0. 16
Assuming L= 6 inches the load at failure for the above
stiffeners is 1,482 pounds as predicted in table 3, u 0.=
1,482/0.053=28,000 lb./ in.2 For an 0.019 sheet of
17ST, P = l.7 X 0.0192.J101X 28,000 = 325 pounds, if
the panel width is greater than b., as it is here. Thi;;
panel would be predicted to carry 2X 1,482+325=3,289
pounds. A test on this size panel gave 3,300 pounds.
A third stiffener added to the above panel half way
between the two shown would result in a predicted load
of 3X 1,482+ 2X 325 = 5,096 pounds. When tested such
a panel carried 5,300 pounds.
A similar panel, 18 inches long, having an 0.033-inch
sheet and four stiffeners of the above type spaced
equally across the sheet, carried 5,600 pounds. The
crushing stress for a length of this stiffener having an
L/p=20 is approximately 28,000 lb./sq. in., ~'.~~~=
28,300, and by using this as the critical stress, we have
18
for C=2, L/p= O.l6 = 112.5, an allowable of 15,500
lb. /sq. in., a load of 15,500X 0.053 or 820 pounds. At
this stress intensity the sheet would carry a load, P=
1. 7X 0.0332-J101x 15,500=730 lb., so the panel would
have been predicted as failing at 4X 820+3X 730=
5,470 lb. instead of the 5,600 pounds shown by test.
This procedure has been checked against a number of
tests with very satisfactory results. It is based on
Gall's and Lundquist's procedures, Gall having been
the pioneer and having assumed the sheet to carry a
stress approximately equal to the yield point of the
material, Lundquist having assumed the stiffener and
effective sheet width tc act as an independent column.
Lundquist's method is described more completely below.
It appears from the foregoing that stiffeners should be
proportioned to develop as high a stress as possible over
the length for which they are to be used so that the
sheets to which they are attached will also carry a high
stress on their effective widths. When a stiffener fails
at low stress int\msities it may in effect be c_onsidered as
21
less than the equivalent of a simply supported edge
for the sheet so the panel will carry less load than is
expected of it.
Anything which improves the effectiveness of the
"edges" of the sheets or increases the stress at which
the sheet stops taking load, naturally improves the
strength-to-weight ratio of a stiffened panel. Anything
which weakens the edge support, such as insufficient
rivets connecting sheet and stiffener, reduces the
strength of the panel.
For joints in which all elements carry compressive
stresses of the same, or approximately the same, mag­nitude
failure is liable to occur by buckling of the out­side
elements between rivets. Tests indicate that when
such buckling occurs the element which buckles be­haves
very nearly the same as a fixed-ended column
so that Euler's formula may be written for the "effec­tive"
column lying between rivets L inches apart. The
formula is
f=C1r2E (;
in which G= 4 for fixed ends and p2= 12/12 for a sheet
of thickness t. Making these substitution:;: and re­writing,
the expression gives
L= l. 814t-JEJJ
This equation is identical with that developed by Mr.
W. L. Howland in his paper "Effect of Rivet Spacing on
Stiffened Thin Sheet in Compression" published in the
October 1936 Journal of the Aeronautical Sciences.
Since it depends upon Euler's formula it cannot be ex­pected
to hold when f exceeds one-half the yield point
stress of the material in hand unless the modulus of
elasticity of the material be reduced so that the value
off given by Euler's formula is the same as that given
by Johnson's parabola, the straight-line or other
standard formula for short columns.
Figure 27 represents the conditions when L is deter­mined
for various aluminum alloys on the basis of E=
10,000,000 lb. per sq. in. for stress intensitives below
20,000 lb. per sq. in. and where it is reduced to accord
with the Johnson parabola for stresses between 20,000
and the yield point. In using the figure enter at the
left with the desired compressive stress and proceed
horizontally to the curve representing the yield point
of the material in hand, thence vertically to the line
representing the sheet thickness. A horizontal line
from that point to the scale at the right gives the rivet
pitch, the distance between the centers of adjacent rivet
holes, required to prevent the sheet from buckling under
the design compressive stress.
There are no known data to corroborate this figure
when the design stress exceeds 20,000 lb. per sq. in.,
but it gives results in reasonable accord with past ex­perience
and current practice and is believed to be
dependable. ]}1r. Rowland's tests appear to check the
curve for stresses of 20,000 lb. per sq. in. and various
tests appear to agree with the results which it gives for
stresses lower than that. Until further data are avail­able
to vindicate or di sprove the basic relations used in
drawing figure 27, it should be looked upon as giving
reasonable results and used with discretion.
22
Design stress in thousands of pounds per .sg. in. -"l'
N
?\JI.._
Rivet pifrh in x"nches ...:.·L°'
FIGURE 27.
Application of the foregoing approximate method to
representative panels tested by Consolidated Aircraft
Corporation and recorded in their report No. 27Z117
gives the following results based on an average yield
point of 42,000, E= 10,000,000. See table 18.
The load on the stiffeners which are of the type shown
in figure 30 was determined analytically and the average
stress on a short length used as the crushing stress
which was then reduced for column effects. The stress
computed for the given L/p of the stiffener was then
employed in determining the effective width of sheet
and the load it carried. Where the actual width was
less than the computed effective width as in the case
of the edge stiffeners, the actual width was of course
the value used. It was assumed that each row of
rivets developed the equivalent of a simply supported
edge so that each stiffener would have, except for over­lapping
areas and discontinuities, two effective widths
of sheet acting in conjunction vvith it.
The normal rivet pitch used on these specimens was
three-fourths inch, but several had 1 inch and a few 1~
inches, staggered. Some reduction in panel load is to
be seen as the pitch is increased, but the results are not
sufficiently uniform to permit definite conclusions.
The differences might be attributed to variations in
thickness of material or in strength p;operties as well
as to changes in rivet pitch. Since few of the speci­mens
failed by buckling between rivets, the only con­clusion
which can be reached is that the stresses de­veloped
were not critical for the pitches used, so the
sheets did not fail between rivets although their strength
was affected to some extent by tho rivet pitch, possibly
as a variation in the elastic support provided the effec­tive
widths of sheet.
TABLE 18
Edge Inter· Load at failure
Stiff- sheet 1 mediate' Num-
Speci- ener Sheet and sheet and berof
menNo. gage gage stiff- stiffener stiff- ener Pre- take eners dieted 1 Test
take
-------- -----------------
240 0.0908 o. 0745 16,530 17,560 3 51,620 52,700
202 .0895 .075 16,530 17, 560 3 51,fi20 52,095
241 .0904 .0735 16,530 17,560 3 51,620 51,090
242 .0707 .072 14,420 15,450 4 59,740 58,450
183 .0713 .0765 14,420 15,450 3 44,290 47,895
198 .0710 .072 14,420 15,450 3 44,290 46,025
243 . 0702 .0745 14,420 15,450 3 44,290 43,755
304 .0888 .0615 14,570 15,450 3 44,590 47,850
204 . 0780 .0620 14,570 15,450 3 44,590 47,630
205 .0875 .0625 14,570 15,450 3 44,590 45,780
246 .0705 .0635 12,460 13,340 4 51,600 53,980
247 .0712 .0652 12,460 13,340 3 38,260 41,170
~1 .0712 . 0651 12,460 13,340 3 38,260 41,720
184 .0647 .063 11, 700 12,580 4 48,560 52,700
199 . 0626 . 0625 11, 700 12,580 3 35,980 38,800
302 .0634 .0640 11,700 12,580 3 35,980 35,140
297 . 0206 . 0195 2,006 2,006 4 8,024 7,980
298 .199 .0197 2,006 2,006 3 6,018 5,860
299 .210 .0194 2,006 2,006 3 6,018 6,355
1 Based on nominal gages of sheet and stiffener, 0.090, 0.072, 0.064, 0.050,
etc.
Lundquist's Method.
In N. A. C. A. Technical Note No. 455, Mr. E. E.
Lundquist proposes the determination of the load on
stiffened sheet by the following procedure:
1. Determine the intensity of stress at which the given
length of stiffener will fail assuming it to act as a simple
column with C-1 for pin-end conditions, C- 2 if th <'
panel is tested with flat ends.
2. Reduce this stress 2 to 5 percent and compute the
effective width of sheet which would carry this r educed
stress if its edges were simply supported.
3. Assume this effective width to act with the stiffener
as a column and determine the location of the centroid
and the values of A, I, and p about an axis through that
centroid and parallel to the plane of the sheet. Since A
will normally increase more rapidly than I, the p of
the combination of sheet and stiffener is generally less
than that of the stiffener alone.
4. Having the LI p of the combination of stiffener and
its effective width of sheet determine the stress at which
it will fail as a column, having due regard to the fixity
coefficient. If this stress is in close agreement with that
computed under 2, the load may be computed by multi­plying
the stress by the area of stiffener and effective
sheet. If it is not in close accord, repeat 2, 3, and 4
until it is. Normally, two or three trials are required.
A formula for determining the variation of stress and
reducing the number of trials bas been developed by
Mr. R. J . White at C. I. T. and is presented in appendix
A of Mr. Sechler's thesis, "The Ultimate Compressive
Strength of Thin Sheet Metal Panels," C. I. T. 1935.
It is,
~=1+[1+~-1~ <To Po _ Ao cl+ ~t.r
o-= stress for stiffener and sheet.
u 0 = stress for stiffener alone.
S = distance from center of sheet to centroid of
stiffener.
p 0 =radius of gyration of stiffener alone.
!=effective width of sheet acting with stiffener.
t=thickness of sheet.
A.=area of stiffener.
As has been indicated in the section on stiffeners, the
addition of a sheet to some stiffener shapes will alter
their crushing strengths materially, hence will modify
the allowable stress for the combination column con­sidered
under the foregoing paragraph (4). When such
an effect occurs allowance should be made for it, al­though
because its effect is normally to increase the
crushing strengths of the sections, it is generally con­servative
to disregard it.
Table 19 gives some comparative results obtained by
applying Gall's original method and Lundquist's
method to panels having stiffeners as shown in figure
26. As may be seen from the table both methods give
satisfactory results for the thin sheet or for the shorter
lengths of panels for the thicker sheet. While Gall's
original method is definitely poor when applied to the
longer panels of heavy sheet, Lundquist's method is
seen to be in good agreement with tests. The approxi­mate
method described above yields results essentially
the same as Lundquist's, although comparative data
are not included in this table. The test data are from
panels tested at M. I. T. as given in the Report on
Aircraft Materials Testing for 1931-32. They were
tested with flat ends, were 11ST, and had average values
153696-40---4
of E=l0,000,000 and yield point=36,000 lb./in.2 All
were 12 inches wide.
As · an illustration of Lundquist's method, let it be
applied to the 12-inch panels having 0.032-inch sheet
in table 19, the stiffeners being similar to those investi­gated
in table 3 except that they are 0.035-inch thick
instead of 0.032. For E= 10,000,000 and u,,,,= 36,000
the effective width of an 0.035-sheet is 0.99 inch which
exceeds the width of back of the channel. Hence, the
back works to the yield point of the material and
carries 0.715 X 0.035 X 36,000= 900 pounds, while each
leg carries 0.385X 107X Oo~::~=342 pounds. The pre­dicted
load for this channel is then 900 + 2X 342=1,584
pounds and the crushing stress 1,584/0.0566 = 28,000
lb./in.2 Substituting this is Johnson's formula, with
L=12 inch, L/μ=12/0.16 = 75, and C=2, gives 22,300
lb.fin. as the allowable stress for the stiffener.
TABLE 19
P redicted loads Num- Test loads
Sheet thick- ber of Lund-
- -------
ness Length stieffresn - mGeathllo'sd quis t.' s method A B
A B ---- -----------
o. orn--o. 020 6 2 3,480 3, 620 3,300 3,100
• 019-- . 020 6 3 5, 410 5, 480 5,000 5,300
. r19- . 020 6 4 7, 340 7,340 6,500 7,100
. 019- . 020 12 2 2, 980 3, 060 2, 890 2,960
. 019- . 020 12 3 4,660 4,630 4,500 3,900
. 019- . 020 12 4 6,400 6,200 6,390 6,470
. 019- . 020 18 2 2,140 2, 120 2, 450 2, 300
. 019- . 020 18 3 3,400 3,210 3,280 3, 270
. 019- . 020 18 4 4, 660 4,300 4, 700 4, 200
. 032- . 033 6 2 4, 250 4, 760 4,400 4,600
. 032- . 033 6 3 6,900 7, 260 6,300 7, 020
. 032- . 033 6 4 9,650 9, 760 9,080 9,500
. 032- .033 12 2 3, 680 3. 77r. 4,190 4,030
. 032- . 033 12 3 6, 060 5,790 6,110 6,025
. 032-. 033 12 4 8,440 7, 810 8,450 7,570
. 032- . 033 18 2 2,910 2,640 2,200 2,700
. 032-. 033 18 3 4, 940 4,050 4, 300 3,830
. 032- . 033 18 4 6,970 5,460 5, 900 5, 300
. 051- . 052 6 2 5, 915 6,500 7,300 6,950
. 051- . 052 6 3 !0, 170 10,640 11, 200 11, 150
. 051- . 052 6 4 14, 780 14,380 14,000 15,000
. 051- . 052 12 2 5, 460 5, 340 5, 250 5,320
. 051- • 052 12 3 9,620 8, 510 9, 720 9,950
. 051- . 052 12 4 13, 780 11, 600 12, 500 13,200
. 051- . 052 18 2 4,620 3, 300 3,350 2,920
. 051- . 052 18 3 8, 360 5,090 4,650 4,890
. 051- .052 18 4 11, 960 6, 880 6, 580 6, 660
Assuming this reduced to 22,000 lb./sq. in. by the
addition of the effective width of sheet, this effective
width is found to be,
b .-- l ·7 X O· 0 32 -Vf 2 21X0 71 03 =1.16 inch. This width may
work with the intermediate stiffeners so they become
"effective" columns as shown in figure 28.
FIGURE 28.
I = 0.00211.
A= 0.0566 + 0.0372= 0.0938 in.2
p= 0.15.
L/p= l2/0.15=80 and the allowable stress is 21,500
lb ./in.2 Then the effective width of sheet is 1. 7X 0.032
f 107 = 1.17 inch which is near enough the assumed -Y 21,500
width, so that p will not be changed, hence 21,500
lb./in.2 can be considered as the ultimate stress for the
"effective" column. Its area is 0.0566 + L17X0.032=
0.0941 inch 2 so it will carry 2,020 pounds.
Since one side of each edge stiffener is at the edge
of the sheet, the entire 1.17 inches of effective width
cannot be assumed to act with the edge stiffeners·
Moreover, the edge of the plate is free to buckle away
from the stiffener, so it will carry 0.385X 101x ~:~;;
0
=
336 pounds or 0.375 X 0.032 X 22,000= 264, whichever is
the smaller; the latter is, so we conclude that the 0.375-
inch on one side of the rivet line and the l.;6
=0.58
inch on the other carry the same stress as the channel.
They form a column having p=0.152, L /p=79 and
the allowable stress is 21,600 lb. /in.2 The edge stiffeners
each carry 21,600 [0.0566 +[(0.375 + 0.58)0.32] = 1,885
pounds.
The panel with the ·two stiffeners would then be
assumed to take 2 X l,885 =3,770 pounds, that with
one intermediate and two edge stiffeners 2 X 1,885 +
2,020 =5,790 pounds, and that with four stiffeners,
2X l ,885+2X 2,020 = 7,810 pounds.
It will be noted that the foregoing, completely_
rational procedure is in reasonable accord with the test
results, being some 8 to 10 percent on the safe side.
At 21,500 lb. /in.2 the stiffener is computed to carry
about 1,220 pounds, whereas, tests of several such
stiffeners show an aver age load of 1,300 pounds for a
12-inch length. Had this been used as the basis of
alllowable stress values for the effective columns, it
would have indicated 23,000 instead of 21,500 lb./in.2
and improved the agreement between predicted and test
results. As data accumulate it is believed satisfactory
agreement can be obtained between predicted and
test results for any normal sheet-stiffener combination,
so that the number of test panels required with any
new construction can be reduced, eventually, to one or
two and possibly eliminated entirely.
A comparison of predicted and test results on panels
made by the Northrop Corporation, as recorded in their
report No. 146, gives for the extruded stiffener section
shown in figure 29.
Area= 0.185,
E = l0,000,000; y . p.=42,000.
Panel No.
XS- 12166___ _ ________ __ ____ 18
X S-12166-30_____ _______ ____ 22. 5
268756 __ ------- - ------ - --- -- 18
268756-L ___ _____ _________ __ 22. 5
Test loads
14,880 14,400, 14, 000.
13, 800 13, 250, 13, 200, 13, 100.
23, 010 24. 950, 26, 200, 23, 930.
21, 390 25, 600, 23, 540, ~4. 750.
24
Panels XS-12166 and XS-12166-30 were assumed to
act as two stiffeners plus the effective area of 0.025
sheet expected to act at a stress of 36,700 lb./in.2 for
the 18-inch specimens, 34,000 for the 22.5-inch speci­mens.
The other two panels had three stiffeners each
with its effective sheet area and, since the edges were
supported, another effective width of sheet due to the
support at the edges.
JR.
8
.064
Ji --i t-·095 1-t::~ FIGt'RE 29.
A further application of Lundquist's procedure to the
Consolidated Aircraft panels from table 18 will now be
made. Both stiffeners and sheet were 24ST having
E= 10,000,000 and an average yield point of 42,000
lb./in .2 Due to variations in gage of sheet and stiffener
the computations are based on nominal dimensions,
0.072 inches for the stiffener, 0.075 for the sheet. The
actual specimens used in the comparison had some-
11·hat thinner material and should have failed at loads
below the predicted. They failed at somewhat higher
loads, however.
FIGURE 30.
Stiffener 3M005.
Area= 0.2105 sq. in.
p= 0.310 in .
On the basis of critical stress being equal to the
yield point of the material, the effective width, b.
/-------rJp = 1.7X t-y
42
X 103= 26.2t. For the stiffener, b.=26.2
X 0.072= 1.885 which exceeds the width of any flat side,
hence, the crushing stress for a short length section is
equal to the yield point.
p= 0.310 for the stiffener alone and for L = 20", L /p
= 64.5.
Assuming C= 2, the allowable st ress at an LI p of 64.5
is 32,500 lb./ in .2 by Johnson's formula.
25
For the effective width of sheet, assuming allowable
stress ·on sheet and stiffener reduced to 32,000, b.= I. 7
X0.075~= 2.25 inches. There should be 2.25
inches effective, 1.125 each side of each line of rivets,
but there is only 1.375 between rivet lines so the total
effective width of sheet 1.375+ 2 (1.1250)= 3.625 on
intermediate stiffeners.
The c. g. of the center stiffener is:
0.2105 X 0.352-3.625 X 0.075X 0.0375
0.2105 + 3.625 X 0.075
0.064
= 0.4825 =0.1325
r = 0.0201 + o.2105x o.2202+ 0.212x o.172= 0.0382.
A =0.4825,p=-=0.282.
L/p=o.;~2=71.0 u.11 0 w=30,500 lb./in.2
This is less than the stress assumed in computing b.
so the effective width of sheet must be revised. I t is
l.7 X 0.075.J
30
~~04=2.325 instead of 2.25. This in­creases
the total effective width to 3.70, but makes no
appreciable change in p or <1all ow·
An intermediate stiffener will therefore carry 30,500
(0.2105+3.70X 0.075) = 14,900 pounds.
Since the sheet does not protrude beyond one edge of the
edge stiffeners, the effective width for an edge stiffener
on the basis of u8 110w= 30,500 is 0.25+ 1.375+ 1.16=2. 79
inches.
c. g. of edge stiffener is:
0.2105 X 0.352-2.79X 0.075 X 0.0375 = 0.0664= 0
0.2105+2.79X 0.075 0.4195 · 158
I = 0.020l + 0.2105X 0.1942+ 0.209X 0.1962
= 0.03605
A=0.4195,p= 0.03605 = 0 293
0.4195 . .
L/ p=20/0.293= 68.3,uallow= 31,100 lb. / in.2
/ 107
b.= l.7X 0.075X -y 31.1 X l03=2.28. Total ac-tive
width is, then 0.25+ 1.375+ 1.14= 2.77
inches, which will not change p or <1allow·
Load per edge stiffener and sheet is
then 31,100(2.77X 0.075 + 0.2105) =31, 100
X 0.4180= 13,000 pounds.
A specimen having two edge and one intermediate
stiffeners should therefore take 2 X 13,000+ 14,900=
40,900 pounds. Consolidated specimen No. 243 took
43,755 pounds. A specimen having two edge and two
intermediate stiffeners should take 2X 13,000 + 2X
14,900 = 55,800 pound , whereas specimen No. 242
carried 58,450 pounds in test.
The agreement between predicted and test values in
the above cases is good and indicates what can be done
toward establishing a rational method for predicting
strengths of panels composed of flat sheets and stiff­eners.
Combinations of corrugated and flat sheet may
be treated in the same manner when subjected to com­pression
parallel to the corrugations. With adequate
rivets in the connection between the two, each line of
rivets becomes the equivalent of a simply supported
edge and permits an effective width of flat sheet to work
to the same stress as the corrugated. There is some
evidence that the frequency of support provided by
the corrugated sheet causes a width greater than
b.= I.7t -Vf E to function, in some cases the entire flat <Icr
sheet seems to be effective, but the data available are
inadequate to establish form factors or similar coeffi­cients
facilitate the evaluation of the elastic support
afforded.
In all of the above applications the areas of sheet be­tween
the edges of the effective widths have been
assumed to carry no load. It has been suggested that
these areas might be considered to work at the stress
K1r2E ( t)z intensity causing buckling, that is, u - 12 (l - μ2) b
where K has the values for a simply supported sheet.
Some test results justify this procedure, but the load is
generally very small and, it is believed, is best neglected.
THEORETICAL ~IETHODS
For a completely rational procedure, the designer is
referred to Timoshenko's "Theory of Elastic Stability,"
article 70, page 371. Due to the complexity of the
procedure it will not be presented here. Where the
stiffeners are equally stiff and equally spaced, the
method may be simplified by replacing the actual sheet
and stiffeners with an orthotropic plate as is discussed
by Timoshenko on page 380. The procedure is partic­ularly
well adapted to corrugated sheets and is dis­cussed
in this report under that heading.
SECTION 10. STIFFENED CURVED SHEETS IN
COMPRESSION
EMPIRICAL )fETHODS
Curved sheets with stifferners may be analyzed by
methods similar to those described for flat sheet, it
being assumed that the effective width of curved sheet
acting with a stiffener is identical with that of a flat
sheet. This assumption is reasonable since the tend­ency
is for the flat sheet to bend near each stiffener
when the load approaches the ultimate, whereas the
less effective part of the sheet between stiffeners buckles
under the load. Hence, by t reating the effective width
of curved sheet as though it were flat and computing
the properties of the "effective" column on that basis,
it is possible to obtain the load on the stiffeners and
adjacent sheet by the Approximate method or by
Lundquist's method.
When the radius of curvature is small the load car­ried
by the areas of sheet between effective widths is
appreciable and should be added to that taken by the
stiffeners. The most satisfactory method for evaluat­ing
these loads is that developed by Sechler and de­scribed
above in the section on compres~ive loads in
curved plates. Sechler recommends the stress on the
areas between effective widths be taken as u= 0.3E -k
and that the load on these areas be added to that com­puted
for the effective widths. Comparisons with test
data show this procedure to be in good agreement with
test data.
26
For preliminary determination of sizes, :figures 11
and 12 are useful in obtaining the loads carried by the
sheet on the assumption that it acts independently of
the stiffener. In using these curves, it is assumed
that the stiffeners suffice to give the equivalent of a
simply supported edge at each point of connection to
the sheet so the width to be used in determining the
coefficients is the width between stiffeners. The
curves were obtained on the basis of tests made on
simply supported curved sheets of l 7ST aluminum
alloy and are in good agreement with the test data.
However, due to the fact that the elastic support of a
stiffener having high L/p is not the equivalent of a
simply supported edge since the stiffener fails before
the effective width of sheet reaches the yield point
stress for the material, these curves indicate higher
loads than can be carried on sheets braced by long or
slender stiffeners. For thin sheets or sheets braced by
stiff members, the agreement is good between loads
predicted by adding stiffener strength to sheet strength
and loads obtained by test. In any case the procedure
being simple and easy to apply will be found helpful in
determining approximate sizes for trial designs.
Since data are available on a series of curved panels
of l 7ST similar to the fiat panels represented in table
19, they will be used to show the degree of approxi­mation
involved in the above methods.
Applying Lundquist's method first we have from
page 23 the loads on 12-inch stiffeners and effective
widths of sheet as 2,020 pounds on an intermediate
stiffener, 1,885 pounds on each edge, the effective
width of 0.032 sheet being 1.17 inches. For the panel
having two stiffeners and a 30-inch radius of curvature,
the width of panel between edges of "effective" widths
would be 12- (2X %+1.17) = 10.08 inches and the
0.032
allowable stress on that area, u=0.3X 107 X ~=
3,200 lb./in.2 The intermediate section would there­fore
carry a load of 10.08X 0.032X 3,200= 1,030 pounds,
so the total for two-edge stiffeners and this section
should have been 2X 1,885+ 1,030= 4,800 pounds,
whereas the test panel carried 4,300. For a 10-inch
radius we would expect the intermediate section to
carry a stress of 9,600 lb./in.2, or a load of 3,090 pounds,
while the edge stiffeners would carry the same 1,885
pounds each. The predicted load would be 6,860, but
the panel carried only 5,420 pounds in test.
Considering similar panels with two edge and one
intermediate stiffener the width of each intermediate
f h t Id b
12- (2X%+2Xl.17)
area o s ee wou e 2 4-46
inches and the load on that with the 30-inch radius
would be 4.46X 0.032X3,200=457, so the total pre­dicted
on the panel would be 2X l,885+2,020+2X457=
6,704 pounds. The test panel carried 5,700 pounds.
With the 10-inch radius the load on the intermediate
areas would be 1,371 pounds each, so the panel should
have taken 2X l,885 + 2,020 + 2X 1,371 = 8,532, whereas,
the test panel failed at 7,400 pounds.
For the panels having two edge and two intermediate
stiffeners, the agreement is considerably better, the
predicted load for the 30-inch radius being 8,602
pounds, the test 8,800, while that predicted for the
10-inch radius is 10,186 and the test 11,100 pounds.
As is obvious the Lundquist-Sechler method is not
in perfect accord with test results, although the errors
involved in the above comparison are larger than
normally occur. Since there is an improvement in the
case where the stiffeners are close together, it is be­lieved
that part of the discrepancy between predicted
and test results is due to the stiffeners not contributing
elastic support to the sheet equivalent to simply
supported edges.
Further studies should be made on this problem
to develop a method in closer accord with test results.
Pending such studies it is probably advisable to neglect
the loads on the intermediate widths of sheet and as­sume
that only the stiffener and its effective width of
sheet carry load.
Table 20 gives a comparison between the approxi­mate
method and test results for the panels just
investigated by the Lundquist-Sechler procedure. The
stiffener is rated at 1,300 pounds on the basis of tests
made on several 12-inch specimens and the sheet loads
are based on that taken by a fiat sheet of the same
thickness times the coefficient K 1 and K2 obtained
from :figures 11 and 12 to provide for the effect of
width, length, and radius/thickness ratio.
It is to be noted that the errors involved in this ap­proximate
method are somewhat, but not much greater
than those for the Lundquist-Sechler system. Neither
method is completely satisfactory for design purposes ,
but they appear to be the best available at present. It
is probable that the Lundquist-Sechler method may be
used for stainless steel and other materials, but com­parisons
made between loads predicted by the approxi­mate
method and those obtained in test indicate too
large discrepancies to permit its application to the
design of members other than aluminum alloy.
TABLE 20
I Number Load on Radius of Width Load on Loads at failure
Skin Load on between
stiff~~ers stiffeners thickness curvature flat sheet rivet Ki I(' curved
(inches) rows sheet Predicted Test
----------------------------· ------
2 2,600 o. 0335 30 1,190 11. 25 2.310 o. 839 2,310 4,910 4,300
I 3 3,900 . 0320 30 1,080 5.63 1. 610 .836 2,905 6,805 5,700
4 5,200 .0320 30 1,080 3. 75 1. 350 .836 3,660 8,860 8,800
2 2,600 . 0335 10 1,190 11.25 4. 650 .870 4,815 7,415 5;420
I 3 3,900 . 0320 10 1,080 5. 63 2. 675 .869 5,020 8,920 7,400
4 5,20() .0320 JO 1,080 3. 75 1. 975 . 869 5,560 JO, 760 11, 100
27
FIGURE 31.
SECTION 11. CYLINDERS IN BENDING
UNSTIFFENED CYLINDERS
Lundquist, in N. A. C. A. Technical Note No. 479,
gives the critical stress on the extreme fiber of a cylinder
subjected to pure bending as <rb=KbE where Kb is a
coefficient depending on the dimensions and imperfec­tions
of the given cylinder . . K. b, plotted against R/t is
given in figure 31, curve A being based on Robertson's
theory for cylinders, curve Bon Southwell's theory and
curve C representing the lower limit of test data. For
design purposes, it is desirable to use the lower, more
conseryative coefficients based on curve C.
The points plotted between curves B and C were
obtained by Mossman and Robinson in their cylinder
tests at Stanford University. They lie approximately
on a curve whose equation is ,r= 0.3E·Jr the expression
suggested by Sechler as representing the stress on the
intermediate areas of simply supported curved panels.
In N. A. C. A. Technical Note No. 523, Lundquist
presents diagrams showing the effect of shear on the
allowable stress in bending on a thin walled cylinder.
Figure 32 presents curves giving the percent of the
allowable stress in pure bending developed for various
ratios of M/RV where Mis the moment, V the shear
at the critical section of radius R. The curves are
given for different ratios of bending stress at the extreme
fiber to shear stress at the neutral axis by being plotted
M "b M V
against RV=~ where "b= 1rR2t and <r ,= 1rRt and.
for various ratios of allowable stress in bending, Sb, to
allowable stress in shear, S., between 0.25 and 10.0.
Sb may be taken as the allowable stress in pure bending
on a cylinder of the same dimensions while S .= 1.25 S.
where S, represents the allowable shear stress on a
cylinder subjected to pure torsion.
13endin9- sfre11s d ta.gra.rn,:
o7""1.): _._ - ~
{ V..l:'_: :c--::;...-,.,,,,-:::1--~:::::: -
t BO I 'II/, ~/'/~. ;;::::::.i;::--:::;:::.- f--"t:::; i--i::::-~ l W{ i~VV ·.~~~ ~-~~~~- -
"' I/ //'!Iv /V.: v,,. ,.,vvr;::-t::,~
b-, Tf/' '/. !/'.:i,:;; I / ,,-t;:. ~t:;:: -S 4-0 I 1,vv;r,-V, ~ :;.:t:'-
"g 20 / 1 (/,t,;~
~ II I/ t;:.-V
0 2 J 4 S 6 tJ 10 l1 t:.
ff/R.V
Chart for bendin.g strength
FIGURE 32.
SECTION 12. STIFFENED CYLINDERS
Fuselage structures are normally built with stiffeners
running longitudinally and with bulkheads or frames
transversely. Many wings have similar structures, the
stiffeners running spanwise and the ribs furnishing the
28
transverse stiffness. The average fuselage is so nearly
circular that it may be treated as a cylinder but the
normal wing covering is of such large radius of curva­ture
that it is generally analyzed as a flat sheet. Both
str.uctures are s·ubjected to bending moments and shear
and each involves problems concerning the allowable
stresses in compression and shear on stiffened sheet.
It is the purpose of this section of this report to investi­gate
static test and other data with a view toward
determining whether or not the failing strength of
complete wing or fuselage structures may be com­puted
by the methods used in previous sections on flat
and curved panels if such methods be modified by the
use of empirical fixity coefficients, arbitrary panel
widths, and similar devices.
Stress distribution studies on stiffened circular
cylinders made at M. I. T. and, on fuselages tested at
Wright Field (see A. C. I. C. 684, An Investigation of
the Stress Distribution Due to Bending and Torque in
the Boeing XP-9 Semi-Monocoque Fuselage) show that
the maximum stress does not occur at the fiber most
distant from the neutral axis unless the transverse
stiffeners are sufficiently close together to prevent any
distortion of the cross section under load. A series of
four circular cylinders, 40 inches in diameter by 10 to
15 feet long, was tested at M. I. T., one having no
transverse frames except at the support and load
points, the others having bulkheads spaced 18, 12, and
6 inches, respectively. These bulkheads were %-inch
fir plywood with 20-inch diameter access holes through
them so there was a 10-inch expanse of plywood in a
radial direction to stiffen the skin at any point. These
bulkheads approximated infinite stiffness for the 0.032
and 0.020 skin used.
Each cylinder had 16 channel-shaped stiffeners
running longitudinally, the spacing being uniform at
22.5° 7.85-inch intervals around the circumference.
Strain gage measurements made at each stiffer:er by
8-inch Berry gages and 1-inch Huggenbergers showed
that the most stressed elements on the compression
side of the cylinder having no transverse bulkheads
were at the stiffeners at the ends of the 45° radii
instead of at the fiber most distant from the neutral
axis, the stress being about 2,850 lb./in.2 at the 45°
stiffener, aE compared with 1,900 at the extreme fiber,
for a ratio of M/1=107.5. With the 18-inch bulkhead
spacing and the same J.11/J ratio, the maximum stress
still occurred at the 45° stiffener, but was practically
the same as that at the 22.5° and extreme fiber stiff­eners,
the variation being between 2,000 and 1,900
lb./in.2 With the 12-inch spacing of bulkheads the
stress at the 22.5° and extreme fiber stiffeners was
practically the same, about 2,300 lb./in. 2, the distribu­tion
across the rest of the cylinder approximating that
from the beam theory. In the case of the 6-inch
spacing the cylinder did not distort appreciably from
its circular section and the stress distribution was very
nearly that from the beam theory, the maximum
variation being about 8 percent at the 22 . .5° stiffener.
The theory indicated 2,100 lb./in.2 at the extreme
fiber and the test showed 2,200 for the J.l!l/ I ratio used.
At higher M/I ra1ios the discrepancy between measured
,;tress and beam theory became larger.
The conclusion is reached, then, that the methods of
analyzing ordinary structures in bending are not
directly apnlicable to stressed-skin fuselages unless the
internal stiffening provided is adequate to prevent
distortion of the cross section. This is difficult to do
in the average structure, although r ecent tests appear
to show that it is less difficult t.han it has been previ­ously
considered.
A second series of three cylinders recently tested at
M. I. T. showed that very flexible rings, when used as
intermediate transverse frames, sufficed to maintain
the circular cross section in the plane of the ring,
although they permitted the cylinder to distort between
frames. These cylinders were identical with the series
mentioned above except that the stiffness of the
transverse frames was varied and that the longirndinal
stiffeners were placed on the outside of the cylinder to
obviate cutting the frames. The frames were spaced
at 12-inch intervals so that, with the cylinder of the
first series having plywood bulkheads at 12-inch spac­ing,
there are four cylinders available for comparison.
The pertinent data are summarized in table 21.
TABLE 21
T ype of transverse frame
%-inch plywood ______ __________ _____ ___ __ _
lY,-inch hat --- - - -- ---- __ __ -- - ----- -- -- -- --
Y,-inch hat __ __ ----- --- -- - -- - -- - ---- ----- --
2% x Yz incb angles--- ------ - -- --- -- -------
I
Momentat
I of frame maximum
section about load on cylin­its
centroid der in inch­pounds
Infinite
0. 0217
. 00491
. 000977
350,000
332,000
329,000
337,000
It appears from the above tests that the stiffness
of the transverse frames has little effect upon the ulti­mate
strength of a circular cylinder carrying bending
loads. The cylinder having a very rigid plywood bulk­head
carried but 5 percent more load than one having
very flexible frames made of two angles, % x J,'2 inch
spaced 1 inch apart and having the }'2-inch legs out­standing.
The 1}'2- and 7~ inch hat-shaped frame mem­bers
were of the square or "high-hat" section, having
%-inch legs, spaced 1 inch apart., attached to the skin
of the cylinder. The heights of the "hats" were 1}'2
and )1i inch respectively.
On the basis of these tests, which are too few in
number to be conclusive, it would seem more desirable
structurally to use a number of light, transverse framef!
at frequent intervals to maintain the cros ,-,ectional
shape rather than a few heavy sections spaced far
apart. It would also appear that sections stiff enough
to withstand handling or accidental loads would s uffice
for intermediate members which carry no external or
concentrated loads.
In discussing these results with Mr. E . E. Lundquist
of the N. A. C. A., he suggested that a tentative criterion,
based on an approximate theoretical analysis, for the
relative strengths of longitudinal and transverse stiffen­ing
members might be taken as
Ir > h
DL Dr
where IT represents the moment of inertia of the
29
transverse stiffener section, h that of the longitudinals;
where D L is the distance along the transverse stiffeners
between longitudinals and where Dr is the distance
along the longitudinals between transYerse stiffeners.
Some criterion of this sort is desirable for use in design
and this is suggested for consideration and for modifica­tion
as subsequent data confirm it or show it to be
incorrect.
The tests on the last series of cylinders showed a
considerable shift of the neutral axis below the hori­zontal
diameter and indicated that the effective section
modulus, l /y, on the compression side of these cylinders
as they approached ultimate load was about half that
of the cylinder based on the properties computed for
the neutral axis at the horizontal diameter. Such a
phenomenon would be expected in view of the reduced
effectiveness of the skin once it had buckled and, as
the buckling is progressive as the load increases, it is
obvious that there will be a change in the neutral axis
with change in load.
This leads us to the following procedure, modified
from a method developed by Walter H. Gale, former
Research Assistant at the M. I. T., which is in fair
agreement with the strength of these cylinders. It is
rational and should be applicable to structures of other
shapes but requires further checking to establish the
degree of error involved in its use.
1. For the first approximation assume the longi­tudinal
stiffeners and the entire skin effective in carry-
12.12
ing stress, except that portions adjacent to cut-outs or
other discontinuities should be omitted when deter­mining
the locus of the neutral axis and the moment
of inertia of the section.
2. Determine the stress at each stiffener point and
at the midpoints of the panels of skin between stiffen-ers
by use of the ordinary beam formulaf= 1~Y.
3. Assuming that some of the skin on the compression
side of the section will buckle under these stresses, and
so change the location of the neutral axis and the values
of I and y, multiply the stresses determined under (2)
by a suitable coefficient for the section considered. (A
limited experience in the application of this method
shows that two is a reasonable factor for circular sec­t
ions.)
4. Determine the effective widths of skin acting
with each stiffener on the basis of these modified stresses
and compute "efficiency factors" for the panels of skin
between these effective ,vidths. The "efficiency factor "
is the ratio of the compressive stress, causing the panel
to buckle, cr=0.3E~, to the stress computed at the mid­point
of the panel under (3).
5. Determine the area of each stiffener with its
effective width of sheet and assume it to act at the cen­troid
of the stiffener. Determine the area of the panels
of skin between the effective widths of sheet acting
with the stiffeners, multiply these areas by the "effi-t
C.G. of tension skin
20
F IGURE 33.
30
ciency factor" computed for the panels under (4) and
assume the resultant effective areas to act as though
concentrated at the midpoints of the panels. This
operation is the equivalent of saying that the skin be­tween
stiffeners acts to carry its normal buckling stress
in compression, but no more.
6. Determine the locus of the neutral axis and re­compute
the properties of the section on the basis of
the effective areas carrying stress in compression, the
whole area in tension.
7. Determine the stress intensities at the extreme
fiber and at any other points which might be critical
and compare them with the allowable stresses at these
points.
8. The allowable compressive stress on the combina­tion
of stiffener and effective sheet area may be ob­tained
as in the case of stiffened panels, using a fixity
coefficient of 1.0 to 1.5 on the stiffener and a column
length equal to the distance between transverse stiffen­ers
or bulkheads. It would seem reasonable that the
column length assumed might be greater than the dis­tance
between transverse members when such members
are flexible bnt the data in hand do not appear to justify
such an assumption, the Yery flexible transverse rings
in conjunction with an 0.020-inch skin having been
adequate to provide the longitudinal stiffeners the sup­port
necessary to develop a fixity coefficient greater
than 1.0. The allowable tensile stress is, of course,
tensile strength of the material.
9. The compressive stresses computed under (7)
should not exceed the allowables determined under
(8). If they do, the overstressed parts should be
assumed to buckle and the section properties be re­computed
on the basis of these parts being only par­tially
effective. While it is sometimes possible to show
a condition of equilibrium to exist with one or two
stiffening members buckled and out of action, it is be­lieved
that such structures are unsafe and that they
should not be used in aircraft.
An application of the method will now be made to
one of the 40-inch diameter cylinders discussed above.
The one having a 12-inch spacing between transverse
frames is chosen because it is representative of. the
spacing used in cylinders of this diameter. The sec­tion
and pertinent dimensions are shown in figures
33 and 34.
Assumed
Stiffener stress,
q"
!__ ___ ______ _____ _______ 20,500
3? ___ -_-_-_-_-_-_-_-_-_--_-_-__-_--__- _- _-_-_-_- 1148,,590000
4____ __________ ___ ____ __ 7,860
5__________ __ ___________ 0
Effective
width I
0. 75
. 78
.89
I. 21
Effective
area
0. 0150
.0156
.0178
. 0242
Stiffener
area
o. 0566
.0566
.0566
. 0566
M=350,000 in.-lb.
The moment of inertia of the entire cross section is:
I= 2,r(0.020) (20) 3 + 16(0.0566)(20)2
502+ 181= 683in.'
2 2
y=20 in. I /y=34.15
On the assumption that the entire cro5s section
carries stress:
f =350,000 X 20=lO 2•0 I 683 'u
350,000 X 18.45
683
f
_350,000 X 14.15
3 - 683
f _350,000X 7.66
· , - 683
fo = O
9,450
7,250
3,930
f A= 35Q,00
6
0si 19.23 = 9,860
fs 350,000X lG.30
683
f
_350,000 X I0.91
c- 683
fo
350,000X3.83
683
8,350
5,600
1,965
For the 0.020 skin the stress at which buckling starts
is o.3Xl~
1
0
X0.020=3,000 lb./in.2 so panels A, B, and
C would be expected to buckle and carry 5tresses not
to exceed 3,000 lb./in.2 Panel A will then be ~·~~~ =
'
0.304 effective; B, ::~~~=0.359, and C, ~'.~~~=0.535.
A limited experience in computing l /y of the effective
section indicates it to be between 50 and 60 percent of
the l/y for the entire section so the stresses on the
stiffeners and panels of skin as computed above will be
doubled. It will be assumed that they are doubled
in the following evaluation of effective widths and
efficiency factors.
Effective
stiffener
area
0. 0716
. 0722
. 0744
.0808
Skin
panel
A n
C
D
Panel Efficiency
width factor
7. 08 0. 152
7. 02 .180
6.80 • 268
7. 25 • 765
Effective
width
1.075
I. 265
1.820
5. 55
Effective
area
o. 0215
.0253
.0364
.1110
The trial neutral axis of the effective section may
now be determined: ·
1. 0.0716 X 20.00 = 1.432
A. 2X .0215X l9.23= .827
2. 2 X .0722 X 18.45 = 2.665
B. 2X .0253X 16.30= .825
3. 2X .0744X 14.15= 2.105
C. 2X .0364X 10.91 = .796
4. 2X .0808X 7.66 = 1.239
D. 2X .111 X 3.83= .851
2: area= 0.9148 10.740
0.0566 (20+36.90 + 28.30+15.32) = 5.690
1r(0.020) (20) (12. 72) = 15.980
-21.67
Neutral a xis lies - 21.67 + 10.74 - 10.93
2.681 1 2.681
= 4.08 inches below diameter 5-13.
1 about neutral axis :
1. 0.0716X 24.082= 41.51
A. .0430X 23.312 = 23.36
2. .1444X 22.532=73.29
B. .0506 X 20.382= 21.01
3. .1488X 18.232= 49.40
C. .0728X 14.992= 16.35
4. .1616X 11.742= 22.27
D. .222 X 7.9I2 = 13.89
5. .1132X 4.082= 1.88
6. .Il32X- 3.582= 1.45
7. .1132X 10.072= 11.48
8. .1132 X 14.372 = 23.38
9. .0566X 15.922= 14.35
Sheet 1.257 X 8.642=93.80
407.51
1 Area=0.9148+ 9 (0.0566) +1.2566=2.681 in.'
..,
~ ~
~ ao ~ \0 N c5
N IN
~ ~ ....
~ .... It') .....
i
t'i
('{
31
I = 1. (of lower skin) + 407.51 = 47.5+407.51 = 455.01
in.•
f = 350,040505X.0 2 4.08 18 , 530 lh . / sq. r. n. a t the ex t reme
fiber.
A second determination of the stresses, neutral axis
and moment of inertia of the effective material will now
be made on the basis of the above approximation.
St resses on stringers and sheet areas :
f
_350,000 X 24.08
,- 455.0 181520
f
_350,000 X 22.53
2 171320
- 455.0
f 3 = 350,040505X.01 8.23=l4 ' 020
f. 350,000X ll.74
455.0 9,030
f _ 350,000 X 4.08
5 - 455.0 3,140
f _350,000X 23.31
A - 455.0 17,920
f _350,000 X 20.38
B - 455.0 15,670
f. 350,000X 14.99 11,530
455.0
f _ 350,000X 7.91
d - 455.0 6,080
Buckling stress on panels=0.3Xl~{0·020 3,000
lb./in.2, as in the first approximation .
. 0724 .0236
' O'[g_'i.
.0267
1 ,QZ4.8.
1
OC)
~ <::::,
co ~
...... t") ~ ~
(.'I') c5 C') ~
~ ~ ~
~ ~ "'-I
FIGURE 34.
Stiffener
1-- ------- -- - -- -- - - --- - -
2--------- .. ----- --- --..
3----------. -------- ----
4--- ------- -- ------- •• --
5----- - --- ---- ---- -- ••••
Stress
18,520
17,320
14,020
9,030
3,140
Effective
width
0. 790
. 816
• 909
1.132
l 1.92
-2-
Effective
sheet area
0. 0158
• 01632
.01818
.0226
.0192
Stiffener
area
32
o. 0566
.0566
.0566
. 0566
.0566
Effective
area
o. 0724
. 0729
.0748
. 0792
.0758
Skin
panel
A
B
C
D
----------
Panel Efficiency Effective Effective
width factor skin width skin area
7. 05 0. 167 1.178 0.0236
6.99 .191 1. 335 . 0267
6.83 . 260 1. 775 .0355
6.33 .494 3.140 .0628
------------ ---- -------- ------------ ------------
1 Effective width of skin with stiffener 5 is that above the Cl;. since all skin below Cl;. is included as a unit.
The neutral axis of the effective section is, then,
1. 1X0.0724=0.0724X24.08= 1.744
A. 2X .0235= .0470X 23.31= 1.096
2. 2X .0729= .1458X22.53= 3.287
B. 2X .0267= .0534X20.38= 1.089
3. 2X .0748= .1496X 18.23= 2.728
C. 2X .0355= .0710X 14.99= 1.065
4. 2X .0792= .1584Xll.74= 1.860
D. 2X .0628= .1256X 7.91= .994
5. 2X .0758= .1516X 4.08= .620
Area= .9745 14.483
9. 1 X 0.0566= .0566 X 15.92= 0.901
8. 2X 0566= .1132X 14.37= 1.628
7. 2X .0566= .1132X 10.07= 1.091
6. 2X .0566= .1132X 3.58= .405
1r(0.020)(20)=1.257 X 8.64=10.857
Area= 1.6532, Mom.= 14.882
New location of the neutral axis is
-14.882+14.483= -0.390=-0 152.
0.9745+ 1.6532 2.627 · m.
below trial location. The I about this axis is, then,
1. 0.0717X 24.232= 42.50
A. .0470X23.462= 25.85
2. .1458X 22.682= 74.95
B. .0534X20.532= 22.53
3. .1496X 18.382= 50.60
C. .0710X 15.142= 16.28
4. .1584 X 11.892= 22.41
D. .1256X 8.062= 8.17
5. .1516X 4.232= 1.70
6. .1132X 3.432= 1.38
7. .1132X 9.922= 11.13
8. .1132X 14.222= 22.88
9. .0566X 15.772= 14.06
Sheet 1.257 X 8.492= 90.50
404.89
l=I0 (of bottom skin)+404.89=47.5+404.89
=452.39 in.2
f 350,000X24.23 18 730 lb/ .
452.39 ' · sq. m.
This is in such close accord with the first approxima­tion
that no further revision is necessary.
The allowable stress on stiffener 1 with its 0.0790-inch
effective width of sheet is found as follows:
0 ~. 1---o. 7'10-:1
o lf-_,o. 16"'0-,
1
flL-:~r-:oo:7±~.,z:-frr.:=-t-=Snr-ix~:-8-°
__1_ o.03~+ LI
FIGURE 35.
Sheet area=0.0158 sq. in.
Stiffener area=0.0566 sq. in.
I. of stiffener=0.00145 in.4
Distance from A-A to c. g. of sheet and stiffener :
(0.020X0.790) (-0.010)
= 0.0158(-0.010) = -0.00